# QED Effect on Black Hole Shadow

Zezhou Hu1, Zhen Zhong2, Peng-Cheng Li1,3, Minyong Guo3∗ and Bin Chen1,3,4

###### Abstract

In this work, taking the QED effect into account, we investigate the shadows
of the static black hole with magnetic monopoles and neutral black holes in
magnetic fields through the numerical backward ray-tracing method. For a
static black holes with magnetic monopole, we obtain the relation between the
shadow radius and the coupling constant. For neutral black holes in the
uniform magnetic fields, we find that the shadow curves deviate very small
from the ellipses for equatorial observer, and we read the linear relation
between the eccentricity and the coupling constant. For $\theta_{o}\neq\pi/2$,
we find that the shadow curves can be well approximated by ellipses in most
cases, except the case that the magnetic field is very strong and the observer
sits around the angle $\theta_{o}=\pi/4$ or $3\pi/4$. Moreover we extend our
investigation to a neutral static black hole surrounded with a current loop.

1Department of Physics and State Key Laboratory of Nuclear Physics and
Technology, Peking University, No.5 Yiheyuan Rd, Beijing 100871, P.R. China

2Department of Physics, Beijing Normal University, Beijing 100875, P. R. China

3Center for High Energy Physics, Peking University, No.5 Yiheyuan Rd, Beijing
100871, P. R. China

4Collaborative Innovation Center of Quantum Matter, No.5 Yiheyuan Rd, Beijing
100871, P. R. China

∗Corresponding author<EMAIL_ADDRESS>
##  1 Introduction

Since the first image of the supermassive black hole in the center of M87∗
galaxy taken by the Event Horizon Telescope (EHT) [1, 2, 3, 4, 5, 6], the
study of black hole physics has entered a new era. The image of black hole not
just provides the direct evidence of the existence of black holes, but also
encodes a lot of information about the black hole and its surroundings. It
opens a new window to investigate various problems in astrophysics and gravity
theories. Thus, how to extract the information from pictures of black holes or
in what ways the various information is stored in the photographs is one of
the most urgent questions that many researchers are working on nowadays.

The black hole shadow may be the primary characteristic of the image of a
black hole. It has been argued that, from the precise shape of the black hole
shadow, many physical parameters related to black holes can be read, such as
the mass, spin and so on [7, 8, 9, 10, 11, 12, 13, 14, 15]. Moreover, it was
suggested in some previous works that wormholes and other ultra-compact
objects (UCOs) could have similar but different shadow curves from a black
hole, which may provide a new observational way to distinguish a black hole
from a wormhole or other UCOs [16, 17, 18, 19, 20, 21, 22, 23]. In addition,
some fundamental problems can be also probed with the help of the study of
black hole shadows as some literature pointed out. These problems include but
are not limited to the extreme environment of black holes, dark matters,
accretion disk, accelerating universe, modified gravity and extra dimensions,
see examples in [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38,
39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57,
58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76,
77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91]. These interesting
questions have greatly stimulated the study of black hole shadows, both
theoretically and experimentally.

Theoretically, the black hole shadows depend largely on our understanding of
photon trajectories in the black hole spacetime. In a vacuum, photons move
along geodesics under the approximation of geometric optics, thus the
equations of motion of photons are completely governed by the background of
the spacetime. However, it is always inevitable that there is some other
fields besides the gravitational field in a real spacetime. Thus, a possibly
realistic scenario is that photons, as a kind of electromagnetic field,
interact with other fields as they travel outside the black hole. In
particular, the magnetic fields play a crucial role in the black hole physics
[92] and there found some evidences showing that magnetic fields may exist
around supermassive black holes. For example, in the vicinity of SgrA* and M87
the intensity of the magnetic field could be as large as 100 G [93, 94].
Therefore, the magnetic field should be taken into account in order to get
closer to the real physical situation. The path of the photon will deviate
from the geodesic in the presence of a magnetic field when considering the
quantum electrodynamics (QED) effects of photons. It was shown that including
the QED effect gives rise to the birefringence phenomenon [95, 96], leading to
the deviation from the geodesics. Consequently, the change of the trajectory
of the photons certainly will change the shape of the black hole shadow
observed by an observer111As for the interaction between photons and other
fields, such as axion-like particles, some interesting works have been made
and some exciting results were found in [97, 98, 99]..

In this paper, we would like to investigate the QED effect on black hole
shadow, due to the background magnetic fields. We will consider three
different magnetic field configurations. We first start with the static black
hole with magnetic monopoles. We find an analytical expression when the
effective coupling between the photon and magnetic field is small. Next we
study carefully the shadow of a Schwarzschild black hole in a magnetic field
by employing the numerical backward ray-tracing method. Moreover, we extend
the study to the case that the black hole is surrounded by a current loop
which generates magnetic field. We find some remarkable features that may help
to determine the mass or magnetic field outside the black hole in the future
EHT experiments.

The plan of our paper is organized as follows: in section 2, we review the
dispersion relations induced by the QED effect. In section 3, we study the
static black holes with magnetic monopoles. In section 4, we investigate the
shadow of a Schwarzschild black hole in a uniform magnetic field at first and
discuss the extension to a Schwarzschild black hole with a current loop. In
section 5, we summarize our results. In addition, we introduce the numerical
backward ray-tracing method carefully in appendices.

In this work, we have set the fundamental constants $c$, $G$ and the vacuum
permittivity $\varepsilon_{0}$ to unity, and we will work in the convention
$(-,+,+,+)$.

##  2 Dispersion relations

In QED, the one-loop vacuum polarization gives rise to the Euler-Heisenberg
effective Lagrangian [100] for the electromagnetic field. For low frequencies,
i.e. $\omega\ll m_{e}$, this is [101]

$L_{eff}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{\mu}{2}\left[\frac{5}{4}(F_{\mu\nu}F^{\mu\nu})^{2}-\frac{7}{2}F_{\mu\nu}F_{\sigma\tau}F^{\mu\sigma}F^{\nu\tau}\right],$
(2.1)

where $\mu=\frac{\hbar e^{4}}{360\pi^{2}m_{e}^{4}}$ with $m_{e}$ being the
electron mass and $e$ being the charge of the electron [102, 103, 104]. Note
that in the geometric unit, $\hbar$ is not necessarily to be set to unity,
instead one finds $\hbar=m_{p}^{2}$ where $m_{p}$ denotes the Planck mass. Let
us consider the case that electromagnetic gauge potential is minimally coupled
to gravity, which leads to the following action

$I=\int\sqrt{-g}d^{4}x\left(\frac{1}{16\pi}R+L_{eff}\right).$ (2.2)

The Einstein’s equation is given by

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi T_{\mu\nu},$ (2.3)

where the energy momentum is

$T_{\mu\nu}=g_{\mu\nu}L_{eff}+F_{\mu\rho}F_{\nu}^{{}^{\,\,\rho}}-\frac{\mu}{2}\left[-5F_{\alpha\beta}F^{\alpha\beta}F_{\mu\rho}F_{\nu}^{{}^{\,\,\rho}}+14F_{\mu}^{\,\,\sigma}F_{\sigma\rho}F^{\rho\tau}F_{\tau\nu}\right],$
(2.4)

and the modified Maxwell equation becomes

$\nabla_{\mu}F^{\mu\nu}+\mu[5\nabla_{\mu}(F_{\alpha\beta}F^{\alpha\beta}F^{\mu\nu})-14\nabla_{\mu}(F^{\mu\sigma}F_{\sigma\tau}F^{\nu\tau})]=0.$
(2.5)

Consider a beam of electromagnetic wave is propagating around the magnetic
black hole, which is equivalent to introducing a small perturbation
$\hat{A}_{\mu}$ into this background. Namely, there is
$A_{\mu}=A^{0}_{\mu}+\hat{A}_{\mu}$, where $A^{0}_{\mu}$ induces the
background magnetic field. Usually, when the wavelength $\lambda$ of the
electromagnetic wave is much smaller than the characteristic scale of the
background, such as the radius $r_{h}$ of the event horizon of the black hole,
i.e. $\lambda/r_{h}\ll 1$, which indeed occurs for celestial bodies including
black holes and neutral stars, then the so-called geometric optics
approximation is valid. Under the geometric optics approximation, the light
lays is described by the following Hamiltonian

$H=(p^{2}-8\mu E^{2})(p^{2}-14\mu E^{2}),$ (2.6)

where

$E^{\mu}=F^{\mu\nu}p_{\nu}.$ (2.7)

From the Hamiltonian one obtains two branches of dispersion relation [95],

$p_{\alpha}p_{\beta}g^{\alpha\beta}+\lambda
F^{\mu\alpha}p_{\alpha}F_{\mu}^{\,\,\beta}p_{\beta}=0,$ (2.8)

with

$\lambda=-8\mu,\mathrm{or}-14\mu.$ (2.9)

From a different perspective, the dispersion relations can be interpreted as
that of the null vector $p_{\mu}$ with the geometry described by the following
effective metric [96]

$G_{\alpha\beta}=g_{\alpha\beta}+X_{\alpha\beta},$ (2.10)

where we have defined a new tensor $X_{\alpha\beta}\equiv\lambda
F^{\mu}_{\,\,\alpha}F_{\mu\beta}$.

##  3 Black hole with magnetic monopole

In this section, let us start from the simplest case: a static black hole with
magnetic monopole. In this case, we ignore the backreaction of the nonlinear
QED correction to the geometry, and the energy momentum tensor is from pure
Maxwell action. Now the stationary Kerr-Newmann is described by the metric

$ds^{2}=-\frac{\Delta}{\Sigma}(dt-a\sin^{2}\theta
d\phi)^{2}+\frac{\sin^{2}\theta}{\Sigma}((r^{2}+a^{2})d\phi-
adt)^{2}+\frac{\Sigma}{\Delta}dr^{2}+\Sigma d\theta^{2},$ (3.1)

where

$\Delta=r^{2}-2Mr+(Q_{e}^{2}+Q_{m}^{2})+a^{2},\quad\Sigma=r^{2}+a^{2}\cos^{2}\theta.$
(3.2)

$M$ and $a$ denotes the mass and angular momentum per unit mass, $Q_{e}$ and
$Q_{m}$ are the electric and magnetic charges. The associated one-form
potential has nonvanishing components

$A_{\mu}dx^{\mu}=\frac{Q_{e}r-Q_{m}a\cos\theta}{\Sigma}dt+\frac{-Q_{e}ar\sin^{2}\theta+Q_{m}(r^{2}+a^{2})\cos\theta}{\Sigma}d\phi.$
(3.3)

For simplicity, we focus on the spherically symmetric black hole, and leave
the case of Kerr in a forthcoming paper [105]. When the black hole is static
and only magnetic charges are considered, the metric can be written as

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),\quad f(r)=1-\frac{2M}{r}+\frac{Q_{m}^{2}}{r^{2}},$ (3.4)

and the gauge field is

$A_{\mu}dx^{\mu}=Q_{m}\cos\theta d\phi.$ (3.5)

In this case, one finds that the effective metric (2.10) is simply given by

$ds^{2}_{\text{eff}}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+h(r)(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),\quad h=r^{2}+\frac{\lambda Q_{m}^{2}}{r^{2}}.$ (3.6)

Since the black hole is static and spherically symmetric, one can always
restrict the photons in the equatorial plane, i.e. $\theta=\pi/2$. The
dispersion relation can be explicitly written as

$-f(r)\dot{t}^{2}+\frac{\dot{r}^{2}}{f(r)}+h\dot{\phi}^{2}=0.$ (3.7)

Moreover, let $p^{\mu}$ denote the 4-momentum of the photons, we use
$q_{\mu}=G_{\mu\nu}p^{\nu}$ to represent the dual vector of $p^{\mu}$ in terms
of the effective spacetime. Then the two conserved quantities, energy and
angular momentum takes the form of

$E=-q_{t},\quad L=q_{\phi},$ (3.8)

which allows us to write the orbit equation as

$\left(\frac{dr}{d\phi}\right)^{2}=V_{eff},$ (3.9)

with the effective potential

$V_{eff}=h^{2}\left(\frac{E^{2}}{L^{2}}-\frac{f}{h}\right).$ (3.10)

By evaluating the equations,

$V_{eff}=0,\quad V_{eff}^{\prime}=0,$ (3.11)

we can obtain the radius of the circular null geodesic, which is determined by

$\frac{f^{\prime}(r_{ph})}{f(r_{ph})}-\frac{h^{\prime}(r_{ph})}{h(r_{ph})}=0.$
(3.12)

In general the solution of this equation is very complicated. However, to
study the effects of QED qualitatively, we assume the the coupling constant
$\mu$ is small. Here and hereafter, we would like to set $M=1$ for simplicity.
As we work in the unit system $c=G=\varepsilon_{0}=1$, $\mu$ has the dimension
of mass square and the magnetic field has the dimension of inverse mass. Their
combination $\mu B^{2}$ is dimensionless, which characterizes the strength of
the QED effect. In the case of black hole with magnetic monopole, by setting
$M=1$ we can find that the smaller black hole, the larger $\mu$. In other
words, the smaller black hole yields the larger $\mu B^{2}$ around the
horizon, such that the nonlinear correction of the QED effect gets enhanced.
In this case, the variation of $\mu$ can reflect the change of the mass of the
black hole. In contrast, for a black hole embedded in a uniform magnetic
field, as we will discuss in next section, the QED effect overall is
independent of the black hole mass.

For a supermassive black hole, the magnetic field around it is small such that
one can neglect the nonlinear QED effect safely. Up to the leading order in
$\mu$, one obtains

$r_{ph}=R_{ph}+\delta\lambda,$ (3.13)

where

$R_{ph}=\frac{1}{2}\left(3+\sqrt{9-8Q_{m}^{2}}\right)$ (3.14)

is the photon sphere radius for the RN black hole with no QED effect and

$\delta=\frac{Q_{m}^{2}\left(R_{ph}-1\right)}{8Q_{m}^{4}+27R_{ph}-18Q_{m}^{2}\left(R_{ph}+1\right)}$
(3.15)

describes the deviation from the result without considering the QED effects.
Due to the spherical symmetry, the photons will fill all the circular orbits
to form a so-called photon sphere. The corresponding constant of motion
$b=L/E$ for this photon sphere is given by

$b_{c}=B_{c}+\Delta\lambda.$ (3.16)

where

$\displaystyle B_{c}$ $\displaystyle=$
$\displaystyle\frac{R_{ph}^{2}}{\sqrt{R_{ph}-Q_{m}^{2}}}$
$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\frac{Q_{m}^{2}}{\sqrt{2\left(R_{ph}-Q_{m}^{2}\right)}\sqrt{8Q_{m}^{4}+54R_{ph}-12Q_{m}^{2}\left(2R_{ph}+3\right)}}$
(3.17)

are the critical impact parameter and the deviation from the result for the RN
black hole without QED effect. The existence of unstable photon sphere means
the appearance of the observable of the black hole, the black hole shadow.
Consider all null geodesics that go from the position of the static observer
at $(t_{o},r_{o},\pi/2,\phi_{o}=0)$ into the past. The critical null geodesics
that orbit around the black hole on the photon sphere will leave the observer
at an angle $\theta$ with respect to the radial line that satisfies

$\tan\theta=\left.\frac{rd\phi}{g_{rr}dr}\right|_{r_{o}}.$ (3.18)

This angle describes the angular size of the shadow of the black hole. For a
static observer at large distance, i.e., $r_{o}\gg 1$, the radius of the black
hole shadow is given by

$r_{sh}=r_{o}\sin\theta=b_{c}.$ (3.19)

Noticing that $\lambda<0$ for both branches of dispersion relation, we
conclude that the radius of the black hole shadow becomes smaller when we
include the QED effect. And the branch corresponding to $\lambda=-14\mu$ has
smaller shadow size compared to the one with $\lambda=-8\mu$.

Figure 1: The shadow of the static and magnetically charged black hole
including the QED effect using the numerical ray-tracing method. We set the
black hole mass $M=1$, the magnetic charge $Q_{m}=0.1$ and the coupling
constant $\mu=1$ with $\lambda=-8\mu$.

In addition to the approximate expressions in Eq. (3.13) and (3.16), it is
possible to study the case that the mass of the black hole is not so large
that the effective coupling constant $\mu\sim 1$ and the effect of QED becomes
significant.222We work in the window that the nonlinear QED effect does not
affect the background geometry but is important for the photon trajectory.
However, since the black hole is spherically symmetric and we can always
choose $\theta=\pi/2$, the variables of the geodesic equations for photons can
be separated. Thus, the radius of the black hole shadow can be calculated
straightforwardly even though Eq. (3.12) is complicated.

Figure 2: The variation of the shadow radius with respect to $\lambda$ for the
static black hole with magnetic monopole including the QED effect. In the
graph of the left, we set the magnetic charge $Q_{m}=0.9$, and $Q_{m}=0.5$ for
the right side. The range of effective coupling $\lambda$ is determined by the
constraint $h>0$ outside the black hole. The values of $\lambda$ is related to
the actual masses of the black holes.

Instead, we give an example of the black hole shadow in Fig. 1 using numerical
backward ray-tracing method. Moreover, we present the variation of the black
hole shadow radius with respect to $\lambda$ in Fig. 2. The causality of the
effective metric requires $\partial_{t}$ is always timelike and the other
translational vectors along the coordinates are spacelike outside the black
hole, that is, $h>0$ has to be hold for $r>2$, which gives us a constraint on
$\lambda$. From Fig. 2, we find $\lambda$ has a wider range of values when
$Q_{m}$ becomes smaller. On the other hand, we find with the increase of
$|\lambda|$, the ratio of the radius of black hole shadow to the event horizon
radius becomes smaller. In addition, it should be pointed out that one may
need consider the backreaction of the nonlinear QED effect when $|\lambda|$ in
Fig. 2 is very large, even though the range of $\lambda$ we consider here is
consistent with $h>0$ under our initial assumptions. Recall that the effective
coupling $\lambda$ is inversely proportional to the mass square of the black
hole, the above fact suggest that for a smaller black hole the shadow radius
is nearer to the black hole horizon. This phenomenon is most obvious for
extreme black hole.

(a) $\Lambda=0.00$

(b) $\Lambda=-0.20$

(c) $\Lambda=-0.40$

(d) $\Lambda=-0.60$

(e) $\Lambda=-0.80$

(f) $\Lambda=-0.95$

Figure 3: The images of the static black hole in uniform magnetic fields. The
inclination angle of the observer is fixed at $\theta_{o}=\pi/2$.

##  4 Neutral black holes in magnetic fields

In this section, we turn to consider a more realistic case that a neutral
black hole is bathed in magnetic fields, at least in the vicinity of the black
hole. Before we get into the specific magnetic field models, let us have some
general discussions. As we know, the magnetic vector is related to
electromagnetic field tensor by

$B^{i}=\varepsilon^{tijk}F_{jk},\quad i=r,\theta,\phi,$ (4.1)

where the repeated index does not mean summation. Here, we consider the case
where there is no electric field, thus we have $A_{t}=0$. Since $A_{\mu}$ is a
gauge field, we can always take the gauge choice $A_{r}=0$. Hence, the general
form of the gauge field reads

$A_{\mu}dx^{\mu}=A_{\theta}(r,\theta)d\theta+A_{\phi}(r,\theta)d\phi.$ (4.2)

where we have assumed the gauge field is independent of $\phi$ for a
stationary spacetime of our interest.

(a) $\theta_{o}=\pi/20$

(b) $\theta_{o}=\pi/4$

(c) $\theta_{o}=\pi/2$

(d) $\theta_{o}=3\pi/4$

(e) $\theta_{o}=19\pi/20$

Figure 4: The images of the static black hole in a uniform magnetic field with
various inclination angles of the observer. $\Lambda$ is fixed at $-0.7$.

###  4.1 Schwarzschild black hole in the uniform magnetic field

In this subsection, we start from a simple case, that is, the gauge filed
$A_{\mu}$ satisfy the source-free Maxwell equations

$\nabla_{\mu}F^{\mu\nu}=0.$ (4.3)

It can be seen that this equation is solved by a gauge potential proportional
to the Killing vector in a Ricci flat spacetime. Namely, a Killing vector in a
vacuum spacetime would generate a solution of source-free Maxwell’s equations.
Thus, for a stationary spacetime of interest, the only non-zero component is
$A_{\phi}$. In fact, this case has been studied by Wald in 1974 [106], and the
solution to the source-free Maxwell equation in a static spacetime was found
to be

$A_{\mu}dx^{\mu}=\frac{B}{2}r^{2}\sin^{2}\theta d\phi,$ (4.4)

where $B$ is the strength of the uniform magnetic field. Due to the presence
of magnetic field, the whole spacetime is not spherically symmetric, instead,
it is axisymmetric. One can expect that in this case, the shadow as seen by a
distant observer is no longer circular anymore, however, this spacetime is
non-spinning, which implies the shadow may not be similar to that of a Kerr
black hole, either. To investigate the shadow of this spacetime is very
interesting.

Figure 5: The outline of the shadow of the static black hole in a uniform
magnetic fields with various inclination angles of the observer. Here
$\Lambda=-0.7$ which is same with that in Fig. 4.

From (2.10) we find that the effective metric for the photon motion is given
by

$\displaystyle ds_{\text{eff}}^{2}$ $\displaystyle=$
$\displaystyle-f(r)dt^{2}+\left(\frac{1}{f(r)}+\lambda
B^{2}\sin^{2}\theta\right)dr^{2}+2\lambda B^{2}r\sin\theta\cos\theta
drd\theta$ $\displaystyle+r^{2}(1+\lambda
B^{2}\cos^{2}\theta)d\theta^{2}+r^{2}\sin^{2}\theta(1+\lambda
B^{2}(f(r)\sin^{2}\theta+\cos^{2}\theta))d\phi^{2},$

where $f(r)=1-\frac{2}{r}$ and $r=2$ is still the coordinate singularity, that
is, the horizon is unchanged for the effective metric. In addition, notice
that $\lambda$ always appears with $B^{2}$, we define a new parameter
$\Lambda\equiv\lambda B^{2}$ for simplicity. The metric components $G_{rr}$,
$G_{\theta\theta}$, $G_{\phi\phi}$ should be positive outside the horizon, we
conclude $0\leq|\Lambda|<1$. Moreover, the effective metric obviously has two
Killing vector, $\partial_{t}$ and $\partial_{\phi}$ such that we can define
two conserved quantities, the energy and angular momentum $L$, as usual

$E=-q_{t},\quad L=q_{\phi}.$ (4.6)

Plugging this into the dispersion relation, we examine the separability of the
Hamiltonian,

$\displaystyle-\frac{E^{2}}{f}+g^{rr}q_{r}^{2}+2g^{r\theta}q_{\theta}q_{r}+g_{\theta\theta}q_{\theta}^{2}+g^{\phi\phi}L^{2}=0.$
(4.7)

Due to the crossing term $g^{r\theta}q_{r}q_{\theta}$ in this equation, the
Hamiltonian cannot be decoupled. In addition, we want to emphasize that this
term cannot be avoided by coordinates transformation, since the effective
metric is noncircular because of the presence of the toroidal magnetic field,
which is in the direction of the rotational Killing vector. In other words,
$A_{\phi}\neq 0$ must lead to the non-zero crossing term $g_{r\theta}$, which
was proved by Carter[107]. Hence, we haves to resort to the “numerical
backward ray-tracing” method to investigate the shadow of such a black hole
inevitably. Here, to make the article concise, we would like to show our
results immediately and leave the method and techniques to the appendix.

In Fig. 3, we show the images of the static and magnetically charged black
hole for various $\Lambda$ when the observer is located on the equatorial
plane. As expected, the shadow is a circle when $\Lambda=0$. When the QED
effects get involved, the shadow is no longer a perfect circle and the
deformation becomes large with the increase of $|\Lambda|$. Moreover, it’s
interesting to investigate the variations of null trajectories and shadow with
the viewing angle since $g_{r\theta}$ exists in the effective metric. In Fig.
4, we show the shadows of the black holes at
$\theta_{o}=\pi/20,\pi/4,\pi/2,3\pi/4,19\pi/20$ by fixing $\Lambda=-0.7$.
Obviously, except $\theta_{o}=\pi/2$, The segment with the longest distance
between two points in the shadow is not horizontal. To get a better idea of
the relationship between the shadow and the observational angle, let us put
these shadows on the same screen in Fig. 5, where the numbers on the
horizontal and vertical coordinates represent the positions of the pixels.
From Fig. 5, we find the profiles are symmetric about the horizontal and
vertical axes for each $\theta_{o}$. In addition, the curve for $\theta_{o}$
and the one for $\pi-\theta_{o}$ coincide. This fact suggests that the uniform
magnetic field does not break the $\mathcal{Z}_{2}$ symmetry of the spacetime.

(a) $\Lambda=-0.10$

(b) $\Lambda=-0.30$

(c) $\Lambda=-0.50$

(d) $\Lambda=-0.70$

(e) $\Lambda=-0.90$

Figure 6: The images of the static black hole in uniform magnetic fields. The
inclination angle of the observer is fixed at $\theta_{o}=\pi/2$. Figure 7:
The variation of the eccentricity with respect to $\Lambda$ for the static and
magentically charged black hole. The inclination angle of the observer is
fixed at $\theta_{o}=\pi/2$.

Another interesting question is whether the closed curve is an ellipse. To
answer this question, let us first focus on the observer’s position on the
equatorial plane. In Fig. 6, we show a series of shadow curves varying with
$\Lambda$. The red curves are strict ellipses whose equations are presented in
Fig. 6. Using our numerical data, we draw the shadow curve with black dots.
One can find that the black dots match the red ellipse very well when
$|\Lambda|$ is not very large, say $|\Lambda|\leq 0.5$. When $\Lambda$ becomes
large, only a few black points deviate from the ellipse a little bit. Thus, we
think the shadow curves seen by an equatorial observer are basically in oval
shape. Moreover, we notice that the ellipse becomes flatter with a increasing
$|\Lambda|$ qualitatively. In order to characterize this relationship more
precisely, we introduce the eccentricity $e$ of an ellipse, and in Fig, 7 we
draw a series of points of $e$ and corresponding $\Lambda$. These data points
fit nicely with a line, showing clearly a linear relation between the
eccentricity and $\Lambda$. With this linear relation, we can infer the value
of $\Lambda$ by measuring the shape of the black hole shadow using the EHT.
Recall that $\Lambda=\lambda B^{2}$, we can further read the magnetic field
strength around the the black hole.

(a) $\theta_{o}=0.1\pi$

(b) $\theta_{o}=0.3\pi$

(c) $\theta_{o}=0.5\pi$

(d) $\theta_{o}=0.7\pi$

(e) $\theta_{o}=0.9\pi$

Figure 8: The shapes of the shadows of the static black hole in uniform
magnetic fields observed by non-equatorial observers with $\Lambda=-0.7$.
Figure 9: The variation of the eccentricity with respect to various
$\theta_{o}$ for the static black hole in a uniform magnetic field with
$\Lambda=-0.7$.

Next, we turn to the shape of the shadow curve observed by non-equatorial
observers. Similarly, by fitting the data to ellipses, we find only a few data
deviate from ellipses when $\theta_{o}$ is near $0$, $\pi/2$, $\pi$, see the
examples in Fig. 8. This is reasonable since the shadow curve nearly circular
when the observer faces up or faces down. For an observer with $\theta_{o}$
being near $\pi/4$ or $3\pi/4$, even though the shape of the shadow looks
still like an ellipsis for $|\Lambda|=0.7$, the deviation from an ellipsis
becomes significant for $|\Lambda|=0.9$. Thus the shadow curves can be
approximately treated as ellipses in most cases. We then show the variation of
the eccentricity with respect to $\theta_{o}$ with $\Lambda=-0.7$ in Fig.9.

###  4.2 A current loop around a Schwarzschild Black hole

In this subsection, we turn to a relatively complicated situation in which
there exists a source on the right hand of the Maxwell equation, that is,

$\nabla_{\mu}F^{\mu\nu}=4\pi J^{\nu}$ (4.8)

Such kind of a solvable model was proposed by J. A. Petterson in 1974 [108].
The current takes in this form

$J^{\phi}=\frac{I}{R_{c}^{2}}\sqrt{1-\frac{2}{R_{c}}}\delta(r-R_{c})\delta(cos\theta)$
(4.9)

where, $R_{c}$ is the radius of the loop, and $I$ is the current density set
to be $1$ later in this paper. Note that the current has axisymmetry and there
is no electric field involved, thus only $A_{\phi}(r,\theta)$ is non-
vanishing. The solution of the Maxwell equation can be expanded in terms of
spherical harmonic functions: for the gauge field in the radius $R_{c}$, it is

$A_{\phi}=\sum_{l=1}^{\infty}\alpha_{l}[rP_{l}(r-1)P_{l}^{1}(\cos\theta)\sin\theta-\frac{r^{2}-2r}{l(l+1)}P_{l,r}(r-1)P_{l}^{1}(\cos\theta)\sin\theta],$
(4.10)

while for the outer gauge field,

$A_{\phi}=\sum_{l=1}^{\infty}\beta_{l}[rQ_{l}(r-1)P_{l}^{1}(\cos\theta)\sin\theta-\frac{r^{2}-2r}{l(l+1)}Q_{l,r}(r-1)P_{l}^{1}(\cos\theta)\sin\theta]$
(4.11)

where $P_{l}$ and $Q_{l}$ are the first and second kind Legendre functions,
and $P_{l,r},Q_{l,r}$ are their derivatives with respect to $r$, $P_{l}^{1}$
is the associated Legendre function. It is worth noting that if all the terms
are included, there appears a divergence when considering the QED effect. To
fix this problem, a simple way is to keep finite orders of harmonics. And for
the simplest case, we can drop all the terms except $l=1$, that is, only the
dipolar term is kept. On the other hand, the continuity of $A_{\phi}$ should
be respected at $r=R_{c}$. In the end, we have

$A_{\phi}=h(r)\sin^{2}\theta$ (4.12)

with

$\displaystyle
h(r)=\begin{cases}-3\pi\sqrt{1-\frac{2}{R_{c}}}V(R_{c})r^{2}&\text{
$2<r<R_{c}$}\\\ -3\pi\sqrt{1-\frac{2}{R_{c}}}V(r)R_{c}^{2}&\text{
$r>R_{c}$},\\\ \end{cases}$ (4.13) $\displaystyle
V(r)=\frac{1}{4}[r+1+\frac{r^{2}}{2}\log(1-\frac{2}{r})].$ (4.14)

Here we ignore the backreaction of the magnetic field to the background.
Moreover, we want to stress that, inside the current loop, the vector
potential can be expressed in terms of a uniform magnetic field $B$ as

$A_{\phi}=\frac{1}{2}Br^{2}\sin^{2}\theta$ (4.15)

with

$B=-6\pi\sqrt{1-\frac{2}{R_{c}}}V(R_{c}).$ (4.16)

However, outside the current loop, the magnetic field $B$ is dependent on $r$,
which takes in this form

$B_{o}(r)=-6\pi\sqrt{1-\frac{2}{R_{c}}}R_{c}^{2}\frac{V(r)}{r^{2}}.$ (4.17)

It is convenient to introduce a new dimensionless quantity

$\displaystyle\kappa(r)=\frac{B(r)}{B}=\begin{cases}1&\text{ $2<r<R_{c}$}\\\
\frac{R_{c}^{2}}{V(R_{c})}\frac{V(r)}{r^{2}}&\text{ $r>R_{c}$},\\\
\end{cases}.$ (4.18)

thus we have $A_{\phi}=\frac{1}{2}\kappa Br^{2}\sin^{2}\theta$ and the
effective metric reads

$\displaystyle ds_{eff}^{2}$ $\displaystyle=$
$\displaystyle-f(r)dt^{2}+\left(\frac{1}{f(r)}+\frac{\Lambda
Z^{2}}{r^{2}\sin^{2}\theta}\right)dr^{2}+\frac{2\Lambda
ZW}{r^{2}\sin^{2}\theta}drd\theta$ $\displaystyle+\left(r^{2}+\frac{\Lambda
W^{2}}{r^{2}\sin^{2}\theta}\right)d\theta^{2}+\left(r^{2}\sin^{2}\theta+\Lambda
fZ^{2}+\frac{\Lambda W^{2}}{r^{2}}\right)d\phi^{2},$

where $\Lambda=\lambda B^{2}$, $Z=\kappa r\sin^{2}\theta$ and $W=\kappa
r^{2}\sin\theta\cos\theta$. Note that the magnetic field $B$ is related to the
position of the current loop and this effective metric is asymptotically flat
which are different from the effective metric discussed in the subsection 4.1.

(a) $R_{c}=3,\Lambda=-0.33$

(b) $R_{c}=3,\Lambda=-0.66$

(c) $R_{c}=4,\Lambda=-0.165$

(d) $R_{c}=4,\Lambda=-0.33$

Figure 10: The inclination angle of the observer is fixed at
$\theta_{o}=\pi/2$. $\delta$ is set to be $0.1$.

Before we move forward to obtain the shadow of the black hole using the
numerical backward ray-tracing method, we need to confine the range of
$\Lambda$ to make sure the spacetime is well-defined outside the horizon.
Because a logarithmic term appears in $V(r)$, we cannot obtain the range of
$\Lambda$ analytically. On the other hand, from Eq. (4.13) , $h(r)$ is
obviously a $C^{0}$ function that will cause discontinuity of Christoffel
symbols of the effective metric. As a result, the null geodesics are
discontinuous at $R_{c}$ in terms of the effective spacetime. To overcome this
difficulty, we employ a seventh-degree polynomial in a small enough interval
$[R_{c},R_{c}+\delta]$ to make $h(r)$ a $C^{3}$ function 333The reason comes
from the fact that the components of the effective metric require $C^{2}$ and
$\partial_{r}h(r)$ (that is, $Z$ and $W$) appear in the metric.

Now, we are ready to apply the numerical backward ray-tracing method to this
effective metric. As in this case, the position of the current loop, the value
of $\Lambda$ and the observing angle all contribute to the shadow of the black
hole, it would be very complicated to consider all the conditions. For the
readability of the article, we leave the full analysis in further study and
list some examples in Fig. 10. It appears that when $|\Lambda|$ becomes large,
the deviation from a circle becomes obvious.

##  5 Summary

In this paper, we studied the QED effects on the shadows of the static black
holes with magnetic monopoles and the static black holes surrounded with
magnetic fields. Firstly, we found that considering QED effect the Hamiltonian
give two branches of dispersion relation which correspond to different
polarizations of light. As a result, with different polarizations, the
observers see black hole shadows of different sizes. This interesting
phenomenon may be observed in future EHT observations.

Next, we moved to investigate the shadows of the static black holes with
magnetic monopoles. We found an analytical expression of the shadow radius
when the effective coupling $|\lambda|\ll 1$ or equivalently the black hole
mass is large. In addition, we gave a numerical study considering all the
allowed $\lambda$’s, and find that the shadow radius is nearer to the black
horizon for smaller black hole due to QED effect.

Then, we turned to consider a more realistic case that a neutral black hole is
bathed in magnetic fields. For a Schwarzschild black hole in a uniform
magnetic field, we obtained the shadow curves using the numerical backward
ray-tracing method. When the observer is located on the equatorial plane, we
found that the shadow fitted nicely with an ellipse for small $|\Lambda|$ and
the deviation from an ellipse was very small even for a vary large
$|\Lambda|$. These facts allowed us to introduce the eccentricity $e$ and
discuss the relation between $e$ and $|\Lambda|$, which turns out to be
linear. For the non-equatorial observers, we found the largest deviation from
an ellipse occurs at $\theta_{o}\sim\pi/4$ or $\theta_{o}\sim 3\pi/4$.
Moreover, we found that the shadow curves for $\theta_{o}$ and
$\pi-\theta_{o}$ overlap since the $\mathcal{Z}_{2}$ symmetry is not broken by
the uniform magnetic field.

Furthermore, we tried to extend our method to a more complicated model in
which there is a current loop around a Schwarzschild black hole. After using
some non-trivial numerical techniques, we found our methods worked well for
this case and showed some numerical results after fixing $\theta_{o}=\pi/2$.
Considering the fact that many parameters are involved in this model, we leave
the complete discussion to the future.

In the present work, we focused on the spherical black holes. It would be
interesting to generalize our study to more realistic astrophysical black
holes with spinning.

## Acknowledgments

The work is in part supported by NSFC Grant No. 11335012, No. 11325522, No.
11735001 and No. 11847241. MG and PCL are also supported by NSFC Grant No.
11947210. And MG is also funded by China Postdoctoral Science Foundation Grant
No. 2019M660278 and 2020T130020. PCL is also funded by China Postdoctoral
Science Foundation Grant No. 2020M670010.

##  Appendix A Backward ray-tracing method

In this appendix, we would like to give a brief but useful introduction on the
numerical backward ray-tracing method which was employed in our study.

Since the effective metric of interest is non-circular and non-rotating in
this manuscript, we assume the effective metric takes the form

$\displaystyle ds_{\text{eff}}^{2}$ $\displaystyle=$ $\displaystyle
g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}+X_{rr}dr^{2}+X_{\theta\theta}d\theta^{2}+X_{\phi\phi}d\phi^{2}+2X_{r\theta}drd\theta$
(A.1) $\displaystyle=G_{\mu\nu}dx^{\mu}dx^{\nu}$

where $g_{\mu\nu}$ is the metric of the background, that is, Schwarzschild
black hole spacetime and $X_{\mu\nu}$ come from the QED effect. The
$4$-momentum vector of a photon is written as

$p^{\mu}=(\dot{t},\dot{r},\dot{\theta},\dot{\phi})$ (A.2)

where the dot denotes the derivative with respect to the affine parameter
$\tau$. By definition, we have the dual vector $p_{\mu}\equiv
g_{\mu\nu}p^{\nu}$, however, $p^{\mu}p_{\mu}\neq 0$ in Schwarzschild black
hole spacetime. Note that $G_{\mu}p^{\mu}p^{\nu}=0$, it is convenient to
define a new dual vector

$q_{\mu}\equiv G_{\mu\nu}p^{\nu}.$ (A.3)

Thus in the effective spacetime, the effective geodesic equations are given by

$\displaystyle\dot{q}_{\mu}=-\frac{\partial H}{\partial
x^{\mu}},\quad\quad\dot{x}^{\mu}=\frac{\partial H}{\partial q_{\mu}}$ (A.4)

in the Hamiltonian canonical formulism, where $H$ is the Hamiltonian of the
null particle, $x^{\mu}$ are $4$-position of the geodesic. Again, the dot
denotes the derivative with respect to the affine parameter $\tau$. Note that
these equations are first-order differential equations with the canonical
variables $(x^{\mu},q_{\mu})$, it is very convenient to do the numerical
geodesic evolution.

Figure 11: In the frame of the observer, the celestial coordinates $\Theta$
and $\Phi$ are introduced to mark each light ray. With the help of the
stereographic projection, a useful map is defined from the celestial sphere to
the screen of our camera.

Next, let’s skip over explaining how the black hole shadow forms in terms of
the null orbits outside the horizon and show the method to determine the
boundary curve of the shadow on the observer’s sky. We assume an observer is
located at $(r_{o},\theta_{o})$ in the coordinates $\\{t,r,\theta,\phi\\}$. It
is worth mentioning that the QED effect has no effect on the observer, thus in
the neighborhood of the observer, a local rest frame can be chosen as

$\displaystyle e_{0}$ $\displaystyle=$ $\displaystyle
e_{(t)}=\left.\frac{\partial_{t}}{\sqrt{-g_{tt}}}\right|_{\left(r_{o},\theta_{o}\right)},$
$\displaystyle e_{1}$ $\displaystyle=$ $\displaystyle-
e_{(r)}=-\left.\frac{\partial_{r}}{\sqrt{g_{rr}}}\right|_{\left(r_{o},\theta_{o}\right)},$
$\displaystyle e_{2}$ $\displaystyle=$ $\displaystyle
e_{(\theta)}=\left.\frac{\partial_{\theta}}{\sqrt{g_{\theta\theta}}}\right|_{\left(r_{o},\theta_{o}\right)},$
$\displaystyle e_{3}$ $\displaystyle=$ $\displaystyle-
e_{(\phi)}=-\left.\frac{\partial_{\phi}}{\sqrt{g_{\phi\phi}}}\right|_{\left(r_{o},\theta_{o}\right)}$
(A.5)

where $g_{\mu\nu}$ is the metric components of the background. Here, we want
to stress that this tetrad is not the only choice, one can choose the right
tetrad according to the actual needs. In a real physical process, the light
comes to the observer from the source, however, we can consider the light
originates from the observer in actual calculation since optical paths are
reversible. A diagram is given in Fig. 11, where the observer is located at
$O$, the red line represents the light and its arrow indicates the direction
of propagation. We use $\overrightarrow{OP}$ to denote the tangent vector of
the null geodesic at the point $O$ in the three-dimensional space. To describe
the position of the photon seen by the observer, it is convenient to introduce
the celestial coordinates as follows. Let’s draw a sphere with $O$ as the
center and $|\overrightarrow{OP}|$ as the radius, the diameter $O^{\prime}S$
is parallel to $e_{1}$ and lies on the equatorial plane, then the line
$OO^{\prime}$ and $OP$ give the angle $\Theta$ which is the first celestial
coordinate of the optical image. In addition, the two lines define a plane,
combining with the sphere, we can identify the great circle that goes through
$P$, $S$ and $O^{\prime}$. Obviously, this great circle intersects the great
circle perpendicular to $e_{1}$ at two points, the point $Q$ is the one closer
to $P$ in Fig. 11. Then vector angle between $\overrightarrow{OQ}$ and $e_{2}$
is the other celestial coordinate, that is, we have complete celestial
coordinates $(\Theta,\Psi)$ hereto. Next, we need move to determine the values
of $(\Theta,\Psi)$. On the one hand, in the frame of the observer, the tangent
vector of the null geodesic takes in this form

$\dot{s}=|\overrightarrow{OP}|\left(-\chi e_{0}+\cos\Theta
e_{1}+\sin\Theta\cos\Psi e_{2}+\sin\Theta\sin\Psi e_{3}\right)$ (A.6)

where the minus sigh guarantees that the tangent vector is pointing to the
past. In addition, the undetermined prefactor $\chi$ is introduced to
guarantee $p^{\mu}$ is a null vector in the effective spacetime. Moreover, the
components from the QED effect can be expressed as

$X_{11}=\frac{X_{rr}}{g_{rr}},\quad
X_{22}=\frac{X_{\theta\theta}}{g_{\theta\theta}},\quad
X_{12}=X_{21}=-\frac{X_{r\theta}}{\sqrt{g_{rr}g_{\theta\theta}}},\quad
X_{11}=\frac{E_{rr}}{g_{rr}},\quad X_{33}=\frac{X_{\phi\phi}}{g_{\phi\phi}},$
(A.7)

in the frame of the observer. Then $p^{\mu}q_{\mu}=0$ gives

$\chi=\left.\sqrt{1+\frac{X_{rr}}{g_{rr}}\cos^{2}\Theta-\frac{X_{r\theta}}{\sqrt{g_{rr}g_{\theta\theta}}}\sin
2\Theta\cos\Psi+\frac{X_{\theta\theta}}{g_{\theta\theta}}\sin^{2}\Theta\cos^{2}\Psi+\frac{X_{\phi\phi}}{g_{\phi\phi}}\sin^{2}\Theta\sin^{2}\Psi}\right|_{(r_{o},\theta_{o})}$
(A.8)

In addition, since the photon trajectory is independent of the photon energy,
let us introduce the energy of the photon in the frame of the camera and set
it to be unity, that is,

$1=E_{camera}\equiv
G_{ab}p^{a}e^{b}_{0}=|\overrightarrow{OP}|\chi=-\left.\frac{E}{\sqrt{-g_{tt}}}\right|_{(r_{o},\theta_{o})}$
(A.9)

On the other hand, from the Eq. (A.2) we also have the tangent vector
expressed in terms of the coordinate bases

$\dot{s}=\dot{t}\partial_{t}+\dot{r}\partial_{r}+\dot{\theta}\partial_{\theta}+\dot{\phi}\partial_{\phi}.$
(A.10)

where $\dot{}=d/d\tau$. By comparing Eq. (A.6) and (A.10), we conclude that
the celestial coordinate of the image is determined, once the values of the
4-momentum of a photon is known. Inversely, if the celestial coordinates of
the image are known, the $4$-momentum can be identified. Thus, combing with
the position of the observer, the initial values $(x^{\mu},q_{\mu})$ are
determined, then evolve the geodesic equations numerically, we can obtain the
path of photons. For some special cases, the explicit expressions exist and
the null geodesics are integrable. For example, the null geodesics are
integrable in Kerr black hole spacetime since there exists the Carter constant
in addition to the mass, angular momentum and energy. On the contrary, when a
spacetime does not possess the Carter constant or one is not able to identify
the hidden symmetry, one has to solve the null geodesic equations numerically
and determine the component values of 4-the momentum as we mentioned at the
beginning of this appendix.

##  Appendix B Camera model and stereographic projection

Figure 12: An illustration of the camera’s screen and pixels, and a standard
Cartesian coordinates with the origin $O^{\prime}$ is set up.

The image of the black hole shadow depends on the camera model. Different
models differ in how a viewing direction is mapped onto the plane. As we know,
the simplest and most natural one is pinhole camera model with the perspective
projection. However, the disadvantage of this model is also obvious that the
field of view is relatively narrow. In view of this, we would like to employ
the stereographic projection in this paper, which is often called fisheye
camera model.

As shown in Fig. 11, we put the optical centre of our camera at point $S$,
then the optical image $P$ on the sphere can be mapped to $P^{\prime}$ on a
plane. In this plane, we can set up standard Cartesian coordinates with the
origin $O^{\prime}$, shown in Fig. 12. Then the coordinates in this plane are
given by

$\displaystyle x_{P^{\prime}}$ $\displaystyle=$
$\displaystyle-2|\overrightarrow{OP}|\tan\frac{\Theta}{2}\sin\Psi$
$\displaystyle y_{P^{\prime}}$ $\displaystyle=$
$\displaystyle-2|\overrightarrow{OP}|\tan\frac{\Theta}{2}\cos\Psi$ (B.1)

Figure 13: The camera’s field of view

Next, let us talk about the camera’s field of view, which is characterized by
the angles in the plane $yO^{\prime}z$ and plane $xO^{\prime}z$, respectively.
For convenience, we might as well take two equal angles. An illustrated field
of view is given in Fig. 13. Then, the length of the square screen is given by

$L=2|\overrightarrow{OP}|\tan\frac{\alpha_{\text{fov}}}{2}$ (B.2)

Let’s suppose we take $n\times n$ pixels on the screen, the length occupied
per pixel is

$\ell=\frac{2|\overrightarrow{OP}|}{n}\tan\frac{\alpha_{\text{fov}}}{2}$ (B.3)

We mark the pixels with $(i,j)$, and set the pixel on the bottom left corner
(the top right corner) to $(1,1)$ ($(n,n)$). Also, $i$ and $j$ are chosen from
$1$ to $n$. Combining the cartesian coordinates of the center point of a
pixel, we have

$\displaystyle x_{P^{\prime}}$ $\displaystyle=$
$\displaystyle\ell\left(i-\frac{n+1}{2}\right)$ $\displaystyle y_{P^{\prime}}$
$\displaystyle=$ $\displaystyle\ell\left(j-\frac{n+1}{2}\right)$ (B.4)

Comparing Eq. (B) with (B), we can establish a relation between $(i,j)$ and
$(\Theta,\Psi)$ as follows

$\displaystyle\tan\Psi$ $\displaystyle=$
$\displaystyle\frac{j-(n+1)/2}{i-(n+1)/2}$ $\displaystyle\tan\frac{\Theta}{2}$
$\displaystyle=$
$\displaystyle\tan\frac{\alpha_{\text{fov}}}{2}\frac{\sqrt{\left[i-(n+1)/2\right]^{2}+\left[j-(n+1)/2\right]^{2}}}{n}$
(B.5)

##  Appendix C Source

Figure 14: An illustration of our spherical light source at infinity.

In our work, we use the extended source to illuminate the system. Our model
can be seen in Fig.14 where we cut the photosphere to reveal its interior. We
divide the ball into grids with the network made of latitude and longitude
lines. The interval for two adjacent longitude or latitude line is $\pi/18$.
To create the image of the black hole, we need to assign an appropriate single
color to each grid. Here, we divided the extended source into four colors.
Then the color of the pixels on the screen can be determined based on the
corresponding source point when these null geodesics originate from the
extended source. And the dark area means the corresponding photons are
captured by the black hole.

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# Stellar Velocity Profiles and Line-Strengths out to Four Effective Radii in
the Early-Type Galaxy NGC 3379

A. Weijmans1, M. Cappellari2, P.T. de Zeeuw3,1, E. Emsellem4,
J. Falcón-Barroso5, H. Kuntschner3, R.M. McDermid6,
R.C.E. van den Bosch7, and G. van de Ven8

###### Abstract

We describe a new technique to measure stellar kinematics and line-strengths
at large radii in nearby galaxies. Using the integral-field spectrograph
SAURON as a ’photon-collector’, we obtain spectra out to four effective radii
($R_{e}$) in the early-type galaxy NGC 3379. By fitting orbit-based models to
the extracted stellar velocity profile, we find that $\sim$ 40% of the total
mass within 5 $R_{e}$ is dark. The measured absorption line-strengths reveal a
radial gradient with constant slope out to 4 $R_{e}$.

1Sterrewacht Leiden, Leiden University, Postbus 9513, 2300 RA Leiden, the
Netherlands<EMAIL_ADDRESS>
2Sub-Department of Astrophysics, University of Oxford, Denys Wilkinson
Building, Keble Road, Oxford OX1 3RH, UK
3ESO, Karl-Schwarzschild-Str 2, 85748 Garching, Germany
4CRAL, University of Lyon, 9 Avenue Charles André, 69230 Saint Genis Laval,
France
5ESTEC, Postbus 299, 2200 AG Noordwijk, the Netherlands
6Gemini Observatory, Northern Operations Centre, 670 N. A’ohoku Place, Hilo,
Hawaii 96720 USA
7McDonald Observatory, University of Texas, Austin, TX 78712, USA
8IAS, Einstein Drive, Princeton, NJ 08540, USA

##  1\. Introduction

Although the presence of dark matter dominated haloes around spiral galaxies
is well established (see e.g. van Albada et al. 1985), little is known about
the dark haloes around early-type galaxies. Large regular H i discs or rings,
whose kinematics are often used to constrain the properties of dark haloes,
are rare in these systems (though see Franx, van Gorkom, & de Zeeuw 1994;
Weijmans et al. 2008), so that we are forced to use other tracers of the
gravitational potential.

Here we use stellar kinematics obtained with the integral-field spectrograph
SAURON (Bacon et al. 2001) at large radii in the elliptical (E1) galaxy NGC
3379 to model its dark halo. This galaxy is of intermediate luminosity
($M_{B}=-20.57$) and has a half-light or effective radius $R_{e}$ of 42
arcsec, which corresponds to 2.1 kpc at its distance of 10.3 Mpc.

##  2\. Method

We centred SAURON at 2.6 and 3.5 $R_{e}$ on both sides of the nucleus of NGC
3379, close to its major axis (see Fig. 1, left panel). A single spectrum of
one lenslet is dominated by noise at these large radii, but adding all spectra
of all lenslets together we obtained in three out of our four fields
sufficient signal-to-noise to measure the stellar absorption line-strengths
and the line-of-sight velocity distribution (LOSVD) up to the fourth Gauss-
Hermite moment $h_{4}$. This last parameter is necessary to break the well-
known mass-anisotropy degeneracy when modeling the dark halo (e.g. Gerhard
1993).

|
---|---
Figure 1.: Left: Positions of our observed fields in NGC 3379. The red boxes
denote each SAURON field-of-view. The skylenslets (red short thick lines) are
aligned with the long side of the SAURON field, at a distance of two
arcminutes. The dashed line denotes the major axis of NGC 3379. The underlying
$V$-band image was obtained with the 1.3-m McGraw-Hill Telescope at MDM
Observatory. Right: LOSVD of NGC 3379 out to 4 $R_{e}$. The black stars are
long-slit data from Statler & Smecker-Hane (1999) and the central red dots are
SAURON observations obtained in the original survey (Emsellem et al. 2004).
The red dots at large radii are our new observations, and double the spatial
extend of the data.

##  3\. Results

We measured the LOSVD using the penalized pixel fitting method (pPXF) by
Cappellari & Emsellem (2004). The resulting LOSVD (Fig. 1, right panel) shows
a smooth continuation of existing stelllar kinematic measurements (Statler &
Smecker-Hane 1999).

We use a Schwarzschild model (van den Bosch et al. 2008; van de Ven, de Zeeuw,
& van den Bosch 2008) to fit our measurements, including the central SAURON
field of the original survey (Emsellem et al. 2004) and the long-slit data of
Statler & Smecker-Hane (1999). The black hole mass and the (nearly
axisymmetric) shape of the stellar distribution of NGC 3379 are taken from van
den Bosch (2008). We model the spherical dark halo with an NFW profile
(Navarro, Frenk, & White 1996) with a concentration $c=10$ (Bullock et al.
2001). Our best-fit model is shown in Fig. 2, and has a total halo mass
$M_{200}$ of $1.0\times 10^{12}$ $M_{\odot}$, which corresponds to 10 times
the total stellar mass of NGC 3379.

We obtained line-strength measurements following the procedures outlined in
Kuntschner et al. (2006). We find that the slope of the line-strength
gradients remains constant out to at least 4 $R_{e}$, although our values of
Fe5015 are not consistent with this trend (Fig. 3). Plotting our measurements
on the stellar population models of Thomas, Maraston, & Bender (2003), we find
that the stellar halo population is old ($\sim$ 12 Gyr) and metal-poor (below
20% solar).

Figure 2.: Best-fit model (solid line) overplotted on datapoints. The blue stars are (symmetrized) long-slit data from Statler & Smecker-Hane (1999) and the red dots are our datapoints (horizontal error bars indicate the width of the SAURON field-of-view). Also shown are a model without a dark halo (dashed line) and a model with a ten times too massive halo (dot-dashed line). These models do not fit the observed dispersion and $h_{4}$ profiles. |
---|---
Figure 3.: Left: Line-strength gradients (from top to bottom: H$\beta$, Fe5015
and Mg $b$, in magnitudes) out to 4 $R_{e}$ in NGC 3379. Black triangles
denote SAURON data from the original survey (Kuntschner et al. 2006), and red
dots are our new measurements. Right: H$\beta$ index against [MgFe50]′,
overplotted on the stellar population models of Thomas et al. (2003). Black
dots show measurements from the SAURON survey, while the coloured dots are
averaged along isophotes (see inset for colour coding). The black filled
squares are our measurements at large radii.

##  4\. Conclusion

We showed that by using SAURON as a ’photon collector’, we can measure both
the stellar velocity profile and absorption line-strengths out to large radii
in early-type galaxies. We presented our measurements of NGC 3379 and modeled
its dark halo. In our best-fit model, 41% of the total mass within 5 $R_{e}$
is dark. We will present more elaborate modeling of the dark halo of NGC 3379
and comparisons with literature values in a forthcoming paper (Weijmans et al.
in preparation), as well as a comparable dataset and analysis for the
elliptical galaxy NGC 821.

### Acknowledgments.

It is a pleasure to thank the organisers for a stimulating and fruitful
conference. We gratefully acknowledge Chris Benn, Eveline van Scherpenzeel,
Richard Wilman and the ING staff for support on La Palma.

The SAURON observations were obtained at the William Herschel Telescope,
operated by the Isaac Newton Group in the Spanish Observatorio del Roque de
los Muchachos of the Instituto de Astrofísica de Canarias.

## References

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  * van den Bosch (2008) van den Bosch, R.C.E. 2008, PhD thesis, Leiden University
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# On Column Competent Matrices and Linear Complementarity Problem

A. Duttaa,1, R. Janab,2, A. K. Dasb
_a Jadavpur University, Kolkata , 700 032, India._
_b Indian Statistical Institute, 203 B. T. Road, Kolkata, 700 108, India._
_1 Email<EMAIL_ADDRESS>
###### Abstract

We revisit the class of column competent matrices and study some matrix
theoretic properties of this class. The local $w$-uniqueness of the solutions
to the linear complementarity problem can be identified by the column
competent matrices. We establish some new results on $w$-uniqueness properties
in connection with column competent matrices. These results are significant in
the context of matrix theory as well as algorithms in operations research. We
prove some results in connection with locally $w$-uniqueness property of
column competent matrices. Finally we establish a connection between column
competent matrices and column adequate matrices with the help of degree
theory.

keywords: Linear complementarity problem Column competent matrices
$W$-uniqueness Column adequate matrices.

11footnotetext: Corresponding author22footnotetext: The author R.Jana
presently working in an integrated steel plant

##  1 Introduction

The $w$-uniqueness property is important in the context of dynamical systems
under smooth unilateral constraints. Xu [18] introduced the column competent
matrices. On uniqueness, quite a large number of results are available in the
literature of operations research. The study of uniqueness property of the
solution is important in the context of the theory of the complementarity
system as well as the method applied for finding the solution. For details see
([15], [11], [3], [10]). Ingleton [6] studied the $w$-uniqueness solutions to
linear complementarity problem in the context of adequate matrices.

The linear complementarity problem can be stated as follows: For $A\in
R^{n\times n}$ and a vector $\,q\,\in\,R^{n},\,$ the linear complementarity
problem denoted as LCP$(q,A)$ finds the solution $w\;\in R^{n}\;$ and $z\;\in
R^{n}\;$ to the following systems

$\displaystyle{w\ -\ Az\ =\ q\ ;\ z\geq 0\ ;w\geq 0}$ (1.1)
$\displaystyle{w^{T}z\ =\ 0}$ (1.2)

or show that there does not exist any $z\;\in R^{n}\;$ and $w\;\in R^{n}\;$
satisfying the system of linear inequalities (1.1) and complementary condition
(1.2).

Pang [14] studied local $z$-uniqueness of solutions of a linear
complementarity problem. The LCP$(q,A)$ has unique $z$-solution for all $q\in
R^{n}$ iff $A$ is a $P$-matrix [1]. The $w$-uniqueness property is identified
by a condition on $A$ related to the notion of sign-reversing. Motivated by
the $w$-uniqueness results, we consider column competent matrices in the
context of LCP$(q,A).$ The sufficient matrices capture many properties of
positive semi definite matrices. The aim of this article is to study some
matrix theoretic properties of this class and establish some new results which
are useful to the solution of the LCP$(q,A).$

The paper is organised as follows. In section 2, we include few related
notations and results. Section 3 presents some new results related to column
competent matrices. We develop several matrix theoretic results of column
competent matrices which are related to solution of linear complementarity
problem. Section 4 provides a conclusion about the article.

##  2 Preliminaries

Here any vector $z\in R^{n}$ is a column vector and $z^{T}$ is the row
transpose of $z.$ We write $z=z^{+}-z^{-}$ where $z^{+}_{i}=\max(0,z_{i})$ and
$z^{-}_{i}=\max(0,-z_{i})$ for any index $i.$ If $A$ is a matrix of order $n,$
$\alpha\subseteq\\{1,2,\cdots,n\\}$ and
${\beta}\subseteq\\{1,2,\cdots,n\\}\setminus\alpha$ then $A_{\alpha\beta}$ is
the submatrix with the rows and columns of $A$ whose indices are in $\alpha$
and $\beta$ respectively. A principal submatrix and a principal minor of $A$
are denoted by $A_{\alpha\alpha}$ and $\det A_{\alpha\alpha}$ respectively.
For $A\in R^{n\times n}$ and $q\in R^{n},$ the feasible set of LCP$(q,A)$ is
defined by FEA$(q,A)$ $=\\{z\in R^{n}:z\geq 0,q+Az\geq 0\\}$ and the solution
set is also defined by SOL$(q,A)$ $=\\{z\in\text{FEA}(q,A):z^{T}(q+Az)=0\\}.$
A $z$-solution, $\tilde{z}$ is called locally unique if $\exists$ a
neighborhood of $\tilde{z}$ within which $\tilde{z}$ is the only $z$-solution.
A $w$-solution, $\tilde{w}=A\tilde{z}+q$, is called locally unique if
$\exists$ a neighborhood of $\tilde{w}$ in which $\tilde{w}$ is the only
$w$-solution. Let $\psi:R^{n}\to R^{n}$ and the kernel of the function $\psi$
is defined by ker $\psi=\\{z\in R^{n}:\psi(z)=0\\}.$ The kernel of a matrix
$A\in R^{n\times n}$ is defined by ker $A$$=\\{z\in R^{n}:Az=0\\}.$ Now we
define the column competent matrix.

######  Definition 2.1.

[18] The matrix $A$ is said to be column competent if $z_{i}(Az)_{i}=0,\ \
i=1,2,\cdots,n\implies Az=0.$

Column competent matrices can be singular or nonsingular matrices. Note that
all singular matrices need not be column competent matrices. Consider $A=$
$\left[\begin{array}[]{rr}1&0\\\ 1&0\\\ \end{array}\right]$ which is a
singular matrix. For any $z=\left[\begin{array}[]{rr}0\\\ k\\\
\end{array}\right],\ k\in R,\ z_{i}(Az)_{i}=0,\ i=1,2$ implies that $Az=0.$
Consider another $A=$ $\left[\begin{array}[]{rr}1&1\\\ 0&0\\\
\end{array}\right].$ It is easy to show that $z_{i}(Az)_{i}=0,\ i=1,2$ does
not imply $Az=0.$ Hence $A$ is not a column competent matrix. Let $A=$
$\left[\begin{array}[]{rrr}1&4&3\\\ 2&1&5\\\ 3&2&0\\\ \end{array}\right],$ for
$z=\left[\begin{array}[]{rrr}0\\\ 0\\\ 1\\\ \end{array}\right]$
$z_{i}(Az)_{i}=0,\ \ i=1,2,3$ but $Az=\left[\begin{array}[]{rrr}3\\\ 5\\\ 0\\\
\end{array}\right]\neq 0.$ Here $A$ is a nonsingular matrix but not a column
competent matrix.

Now we define $\psi:R^{n}\to R^{n}$ where $\psi(z)=z*(Az)$ and $z*(Az)$ is the
Hadamard product defined by $(z*(Az))_{i}=z_{i}*(Az)_{i},\ \forall\ i.$ Note
that the product is associative, distributive and commutative.

######  Definition 2.2.

[18] In view of Hadamard product, a matrix $A$ is said to be column competent
if $\text{ker}\ \psi=\text{ker\ }A.$

Column adequate matrices are related to column competent matrices. We start
with definition of column adequate matrices.

######  Definition 2.3.

[1] The matrix $A$ is said to be column adequate if $z_{i}(Az)_{i}\leq 0,\
i=1,2,\cdots,n\implies Az=0.$

We state the following lemma and theorems which are useful for the subsequent
sections.

######  Lemma 2.1.

[18] The matrix $A$ is said to be non-degenerate if and only if $\text{ker}\
\psi=\\{0\\}.$

######  Theorem 2.1.

[18] The following statements are equivalent.

  1. (i)

$A$ is column competent.

  2. (ii)

For all vector $q,$ the LCP$(q,A)$ has a finite number (possibly zero) of
$w$-solutions.

  3. (iii)

For all vector $q,$ any $w$-solution of the LCP$(q,A),$ if it exists, must be
locally $w$-unique.

######  Theorem 2.2.

[18] The following statements are equivalent.

  1. (i)

(a) $A$ is column competent.
(b) $A$ is a $P_{0}$-matrix.

  2. (ii)

$A$ is column adequate.

######  Theorem 2.3.

[1] Let $A\in R^{n\times n}$ be a $E_{0}$-matrix. Then the following
statements are equivalent.

  1. (i)

$A\in R_{0}.$

  2. (ii)

$A\in R.$

We say that $A\in R^{n\times n}$ is a

$-$ $Q$-matrix if for every $q\in R^{n},$ LCP$(q,A)$ has a solution.
$-$ $Q_{0}$-matrix if for any $q\in R^{n},$ feasibility implies solvability.
$-$ $P$-matrix if for each vector $z\neq 0$ there exists an index $i$ such
that $\max_{(z_{i}\neq 0)}z_{i}(Az)_{i}>0.$
$-$ $P_{0}$-matrix if for each vector $z\neq 0$ there exists an index $i$ such
that $\max_{(z_{i}\neq 0)}z_{i}(Az)_{i}\geq 0.$
$-$ $R_{0}$-matrix if LCP$(0,A)$ has unique solution.
$-$ principally non-degenerate if it has no principal submatrix which has
determinant zero.
For further details about the matrix classes in linear complementarity problem
see ([4],[9], [13], [7], [8]).

The principal pivot transform (PPT) has an important role in the study of
matrix classes and linear complementarity problem. The principal pivot
transform of $A\in R^{n\times n}$ with real entries, with respect to
$\alpha\subseteq\\{1,2,\ldots,n\\}$ is defined as the matrix given by

$\displaystyle{A^{\prime}=\left[\begin{array}[]{cc}A^{{}^{\prime}}_{\alpha\alpha}&A^{{}^{\prime}}_{\alpha\bar{\alpha}}\\\
A^{{}^{\prime}}_{\bar{\alpha}\alpha}&A^{{}^{\prime}}_{\bar{\alpha}\bar{\alpha}}\end{array}\right]}$

where
$A^{{}^{\prime}}_{\alpha\alpha}=(A_{\alpha\alpha})^{-1},\;A^{{}^{\prime}}_{\alpha\bar{\alpha}}$=$-(A_{\alpha\alpha})^{-1}A_{\alpha\bar{\alpha}},\;\,$$A^{{}^{\prime}}_{\bar{\alpha}\alpha}=A_{\bar{\alpha}\alpha}(A_{\alpha\alpha})^{-1}$
and
$A^{{}^{\prime}}_{\bar{\alpha}\bar{\alpha}}=A_{\bar{\alpha}\bar{\alpha}}-A_{\bar{\alpha}\alpha}(A_{\alpha\alpha})^{-1}A_{\alpha\bar{\alpha}}.$

Here PPT is only identified with respect to those $\alpha$ for which $\det
A_{\alpha\alpha}\neq 0.$ When $\alpha=\emptyset$, by convention $\det
A_{\alpha\alpha}=1$ and $A^{\prime}=A.$ Here
$A^{{}^{\prime}}_{\bar{\alpha}\bar{\alpha}}$ is said to be Schur complement of
$A.$ We denote the PPT of A as $A^{{}^{\prime}}=\mathcal{P}_{\alpha}(A).$ The
schur complement of $A_{\alpha\alpha}$ in
$A=\left[\begin{array}[]{cc}A_{\alpha\alpha}&A_{\alpha\bar{\alpha}}\\\
A_{\bar{\alpha}\alpha}&A_{\bar{\alpha}\bar{\alpha}}\end{array}\right]$ is a
principal submatrix of the principal pivot transform $A^{\prime}$. For details
of PPT see ([11], [12], [2]).

We establish a connection between competent matrices and adequate matrices
using degree theoretic approach. We provide a brief details about degree
theory in the subsequent section.

###  2.1 Degree theory

Let $f_{A}:R^{n}\rightarrow R^{n}$ be a piecewise linear mapping for a given
matrix $A\in R^{n\times n}$ defined as $f_{A}(e_{i})=e_{i}$ and
$f_{A}(-e_{i})=-Ae_{i}\ \forall\ i.$ We write for any $z\in R^{n},$

$f_{A}(z)=z^{+}-Az^{-}.$

For details see [10]. It is clear that LCP$(q,A)$ is equivalent to find a
vector $z\in R^{n}$ such that $f_{A}(z)=q.$ If $z$ belongs to the interior of
some orthants of $R^{n}$ and $\det A_{\alpha\alpha}\neq 0$ where
$\alpha=\\{i:z_{i}<0\\},$ then the index of $f_{A}(z)$ at $z$ is well defined
and can be written as

$\text{ind}f_{A}(q,z)=sgn(\det A_{\alpha\alpha}).$

Note that the cardinality of $f_{A}^{-1}(q)$ denotes the number of solutions
of LCP$(q,A).$ Particularly, if $q$ is non-degenerate with respect to $A,$
each index of $f_{A}$ is well defined and we can define local degree of $A$ at
$q.$ It can be denoted as deg${}_{A}(q).$ For details see ([1], chapter 6). We
state the following theorem from [10], which will be required to prove one of
our result.

######  Theorem 2.4.

Let $A\in$$R^{n\times n}.$ Let $K(A)$ denote the union of all the facets of
the complementary cones of $(I,{}-A)$. Consider $q\in R^{n}\setminus k(A)$
where $q$ is non-degenerate with respect to $A.$ Let
$\beta\subseteq\\{1,2,\cdots,n\\}$ be such that $\det A_{\beta\beta}\neq 0.$
Suppose $A^{\prime}$ is a PPT of $A$ with respect to $\beta.$ Then
$\text{deg}_{A^{\prime}}(q^{\prime})=sgn(\det
A_{\beta\beta})\cdot\text{deg}_{A}(q).$

##  3 Results on Column Competent Matrices

Hadamard product is important to characterize the complementary condition.
Here we show that the property of column competent matrix is related to
Hadamard product.

######  Theorem 3.1.

Suppose $A$ is a column competent matrix and the function $\psi:R^{n}\to
R^{n}$ defined by $\psi(z)=z*(Az)$ where $z*(Az)$ is the Hadamard product.
Then $\text{ker}\ \psi=\text{ker}A.$

###### Proof.

Let $A$ be a column competent matrix. Then for a vector $z\in
R^{n},z_{i}(Az)_{i}=0,\ i=1,2,\cdots,n\implies Az=0.$ Hence $z\in\text{ker}\
\psi$ implies $z\in\text{ker}A.$ So we write $\text{ker}\
\psi\subseteq\text{ker}A.$ Again by definition
$\text{ker}A\subseteq\text{ker}\ \psi.$ Therefore $\text{ker}\
\psi=\text{ker}A.$ ∎

The following result provides a characterization of non-degenerate column
competent matrices.

######  Theorem 3.2.

Let $A\in R^{n\times n}$ be a non-degenerate column competent matrix. Then
$A\in R_{0}.$

###### Proof.

Let $A$ be a non-degenerate column competent matrix. By Lemma 2.1,
$\text{ker}\ \psi=\\{0\\}$ where $\psi(z)=z*Az.$ By Theorem 3.1, we can write
$\text{ker}\ \psi=\text{ker\ }A=\\{0\\}.$ Let $z$ be the solution of
LCP$(0,A).$ Then $z_{i}(Az)_{i}=0,\ i=1,2,\cdots,n.$ This implies that $Az=0.$
Hence $z=0.$ Therefore, LCP$(0,A)$ has only one solution zero. Hence $A$ is a
$R_{0}$\- matrix. ∎

Note that column competent matrix need not be a $P_{0}$\- matrix in general.
Consider the matrix $A=$ $\left[\begin{array}[]{rr}2&1\\\ 1&-1\\\
\end{array}\right].$ We show that $A$ is a column competent matrix but not a
$P_{0}$-matrix. Now we establish the following result.

######  Theorem 3.3.

Suppose $A$ is a column competent matrix with $A\in P_{0}.$ Then for $0\neq
z\geq 0,$ $(z,0)$ is the solution of LCP$(0,A).$

###### Proof.

Let $A\in R^{n\times n}$ be a column competent matrix with $A\in P_{0}.$ Then
for each $0\neq z,$ $\max_{i}z_{i}(Az)_{i}\geq 0,z_{i}\neq 0.$ If
$z_{i}(Az)_{i}=0,\ i=1,2,\cdots,n$ implies that $Az=0.$ Then $(z,0),z\geq 0$
is the solution of LCP$(0,A).$ ∎

Now we consider the matrix $A=$ $\left[\begin{array}[]{rr}2&-1\\\ -4&2\\\
\end{array}\right]$ is column competent as well as $P_{0}$ and
$(\left[\begin{array}[]{rr}1\\\ 2\\\
\end{array}\right],\left[\begin{array}[]{rr}0\\\ 0\\\ \end{array}\right])$ is
a solution of LCP$(0,A).$ Note that this can be explained using the Theorem
3.3.

######  Theorem 3.4.

Let $A$ be a column competent matrix. Suppose $z\geq 0$ and $z_{i}(Az)_{i}=0,\
i=1,2,\cdots,n.$ Then LCP$(0,A)$ has the solution $(z,0).$

###### Proof.

Since $A$ is a column competent matrix, then for $z\geq 0$ and
$z_{i}(Az)_{i}=0,\ i=1,2,\cdots,n.$ This implies that $Az=0.$ Therefore
$(z,0)$ is the solution of LCP$(0,A).$ ∎

Xu [18] showed that if $A$ is a column competent matrix then $DAD^{T}$ is a
column competent matrix where $D$ is a diagonal matrix. In the next theorem,
we prove that column competent matrices with some additional assumptions are
invariant under principal rearrangement. For any principal submatrix
$A_{\alpha\alpha}$ of $A,$it is possible to rearrange principally the rows and
columns of $A$ in such a way that $A_{\alpha\alpha}$ becomes a leading
principal submatrix in the rearranged matrix $PAP^{T}.$

######  Theorem 3.5.

Suppose $A$ is a column competent matrix. If for any $z\in R^{n},$ either
$z_{i}(Az)_{i}\geq 0$ or $z_{i}(Az)_{i}\leq 0$ for all $i,$ then $PAP^{T}$ is
also column competent where $P$ is a permutation matrix.

###### Proof.

Let for any $z\in R^{n},$ $y=Pz.$ Consider $y_{i}(PAP^{T}y)_{i}=0$ for all
$i.$ This implies that $(Pz)_{i}(PAP^{T}Pz)_{i}=0$ for all $i.$ We know that

$z^{T}P^{T}PAP^{T}Pz=$ $\sum_{i=1}^{n}(Pz)_{i}(PAP^{T}Pz)_{i}$ $=0.$

Hence $z^{T}Az=0$ as $P^{T}P=I.$ We write $\sum_{i=1}^{n}z_{i}(Az)_{i}=0.$ It
means $z_{i}(Az)_{i}=0,\ i=1,2,\cdots,n.$ As $A$ is a column competent matrix,
$Az=0.$ Therefore $AP^{T}Pz=0.$ Hence $(PAP^{T})(Pz)=0.$ Hence $PAP^{T}$ is
column competent. ∎

######  Theorem 3.6.

Let $A$ be a $Z$\- matrix. Suppose $z_{i}(Az)_{i}=0$ for all $i,$ $A|z|\geq 0$
and $Az\leq 0.$ Then $A$ is a column competent matrix.

###### Proof.

Suppose $A$ is a $Z$\- matrix. Consider $z_{i}(Az)_{i}=0$ for all $i,$
$A|z|\geq 0$ and $Az\leq 0.$ As $A$ is a $Z$\- matrix, $Az\geq A|z|\geq 0.$
Now this implies that $Az=0.$ Therefore $A$ is a column competent matrix. ∎

Consider $A=$ $\left[\begin{array}[]{rrr}1&1&4\\\ 2&2&5\\\ 3&4&1\\\
\end{array}\right].$ Note that $A$ is a $R_{0}$-matrix. Now for
$z=\left[\begin{array}[]{rrr}1\\\ -1\\\ 0\\\ \end{array}\right],$
$z_{i}(Az)_{i}=0,\ i=1,2,3$ but $Az\neq 0.$ Hence $A$ is not a column
competent matrix. The class of non-degenerate matrices play an important role
to characterize certain uniqueness properties of the solutions of LCP$(q,A).$
We prove the following theorem to establish the relation between principally
non-degenerate matrices and column competent matrices.

######  Theorem 3.7.

Let $A$ be a principally nondegenerate matrix. Then $A$ is column competent.

###### Proof.

Let $A$ be a principally non-degenerate matrix. Assume that $A$ is not a
column competent matrix. Hence $\exists$ a $0\neq z\in R^{n}$ such that
$z_{i}(Az)_{i}=0,\ i=1,2,\cdots,n$ but $Az\neq 0.$ Without loss of generality,
consider $z=\left[\begin{array}[]{rr}z_{\alpha}\\\ z_{\beta}\\\
\end{array}\right]\neq 0$ where $z_{\alpha}\neq 0,z_{\beta}=0$ and
$A=\left[\begin{array}[]{rr}A_{\alpha\alpha}&A_{\alpha\beta}\\\
A_{\beta\alpha}&A_{\beta\beta}\\\ \end{array}\right].$ Then we consider the
following cases:

case1: Let $\alpha=\\{1,2,\cdots,n\\}$ and $\beta=\emptyset.$ Then
$z=z_{\alpha}$ and $z_{i}(Az)_{i}=0,\ i\in\alpha$. It implies $Az=0,$
contradicts the fact that $Az\neq 0.$

case2: Let $\alpha\subset\\{1,2,\cdots,n\\}$ and
$\beta=\\{1,2,\cdots,n\\}\setminus\alpha.$ Consider
$(z_{\alpha})_{i}(A_{\alpha\alpha}z_{\alpha})_{i}=0,\ i\in\alpha.$ This
implies $A_{\alpha\alpha}z_{\alpha}=0.$ As $z_{\alpha}\neq 0,$
$A_{\alpha\alpha}$ is a singular matrix. It contradicts that the matrix $A$ is
a principally non-degenerate matrix.

Therefore $A$ is a column competent matrix. ∎

Here we consider $A=$ $\left[\begin{array}[]{rrr}3&-2&0\\\ -2&1&1\\\ -3&2&0\\\
\end{array}\right].$ For $z=\left[\begin{array}[]{rrr}2k\\\ 3k\\\ k\\\
\end{array}\right],k\in R,\ z_{i}(Az)_{i}=0,\ i=1,2,3$ implies that $Az=0.$
Hence $A$ is a column competent matrix. However $A$ is neither an adequate
matrix nor a sufficient matrix. For details of sufficient matrices see ([17],
[16], [5]). Now we develop a necessary and sufficient condition for column
competent matrices.

######  Theorem 3.8.

Let $A\in R^{n\times n}$. The following two statements are equivalent:

  1. (a)

$A$ is column competent.

  2. (b)

For $0\neq z=$ $\left[\begin{array}[]{rr}z_{\alpha}\\\ z_{\beta}\\\
\end{array}\right]\geq 0$ with $z_{\beta}=0$ and the submatrix
$A_{\alpha\alpha}$ is singular with $A_{\alpha\alpha}z_{\alpha}=0$ where
$\alpha\cup\beta=\\{1,2,\cdots,n\\}$ and $\alpha\cap\beta=\emptyset,$ the
system

$\left[\begin{array}[]{rr}A_{\alpha\alpha}&A_{\alpha\beta}\\\
A_{\beta\alpha}&A_{\beta\beta}\\\
\end{array}\right]\left[\begin{array}[]{rr}z_{\alpha}\\\ z_{\beta}\\\
\end{array}\right]\neq 0$ (3.1)

has no solution.

###### Proof.

$(a)\implies(b).$ Suppose $A$ is column competent and Equation 3.1 is
consistent. Let $\left[\begin{array}[]{rr}z_{\alpha}\\\ z_{\beta}\\\
\end{array}\right]$ satisfies the Equation 3.1 where $z_{\beta}=0$ and the
submatrix $A_{\alpha\alpha}$ is singular with $A_{\alpha\alpha}z_{\alpha}=0$
where $\alpha\cup\beta=\\{1,2,\cdots,n\\}$ and $\alpha\cap\beta=\emptyset$.
Here $(z_{\alpha})_{i}(A_{\alpha\alpha}z_{\alpha})_{i}=0$ and
$(z_{\beta})_{i}(A_{\beta\alpha}z_{\alpha})_{i}=0.$ But
$\left[\begin{array}[]{rr}z_{\alpha}\\\ z_{\beta}\\\ \end{array}\right]$
satisfies 3.1 which contradicts that $A$ is column competent.

$(b)\implies(a).$ Conversely, let $x\in R^{n}$ be a vector such that
$x_{i}(Ax)_{i}=0$ for all $i.$ Consider $A$ is not a column competent matrix.
Suppose $x_{\alpha}=z_{\alpha}$ and $x_{\beta}=z_{\beta}$ with $0\neq z=$
$\left[\begin{array}[]{rr}z_{\alpha}\\\ z_{\beta}\\\ \end{array}\right]\geq
0,$ $z_{\beta}=0$ and the submatrix $A_{\alpha\alpha}$ is singular with
$A_{\alpha\alpha}z_{\alpha}=0$ where $\alpha\cup\beta=\\{1,2,\cdots,n\\}$ and
$\alpha\cap\beta=\emptyset.$ But the system 3.1 has no solution, i.e. $x$ does
not satisfy Equation 3.1. Therefore $A$ is column competent. ∎

Now we prove the following sufficient condition related to the PPT of column
competent matrices.

######  Theorem 3.9.

Let $A_{\alpha\alpha}$ and the Schur complement $A/A_{\alpha\alpha}$ be
nonsingular of the square matrix
$A=\left[\begin{array}[]{rr}A_{\alpha\alpha}&A_{\alpha\beta}\\\
A_{\beta\alpha}&A_{\beta\beta}\\\ \end{array}\right]$ where
$\alpha\cup\beta=\\{1,2,\cdots,n\\}$ and $\alpha\cap\beta=\emptyset.$ If $A$
is column competent, then $A^{\prime}=\mathcal{P}_{\alpha}(A)$ is column
competent.

###### Proof.

Let $w=A^{\prime}z$ and $z*w=0$ where $*$ is the Hadamard product. Thus we
write

$\left[\begin{array}[]{rr}w_{\alpha}\\\ w_{\beta}\\\
\end{array}\right]=\left[\begin{array}[]{rr}A^{\prime}_{\alpha\alpha}&A^{\prime}_{\alpha\beta}\\\
A^{\prime}_{\beta\alpha}&A^{\prime}_{\beta\beta}\\\
\end{array}\right]\left[\begin{array}[]{rr}z_{\alpha}\\\ z_{\beta}\\\
\end{array}\right].$ (3.2)

The condition $z*w=0$ means $\left[\begin{array}[]{rr}z_{\alpha}\\\
z_{\beta}\\\ \end{array}\right]*\left[\begin{array}[]{rr}w_{\alpha}\\\
w_{\beta}\\\
\end{array}\right]=\left[\begin{array}[]{rr}w_{\alpha}*z_{\alpha}\\\
w_{\beta}*z_{\beta}\\\ \end{array}\right]=0.$ Since
$A^{\prime}=\mathcal{P}_{\alpha}(A),$ we have

$\left[\begin{array}[]{rr}z_{\alpha}\\\ w_{\beta}\\\
\end{array}\right]=\left[\begin{array}[]{rr}A_{\alpha\alpha}&A_{\alpha\beta}\\\
A_{\beta\alpha}&A_{\beta\beta}\\\
\end{array}\right]\left[\begin{array}[]{rr}w_{\alpha}\\\ z_{\beta}\\\
\end{array}\right].$ (3.3)

The matrix $A$ is column competent implies that
$\left[\begin{array}[]{rr}A_{\alpha\alpha}&A_{\alpha\beta}\\\
A_{\beta\alpha}&A_{\beta\beta}\\\
\end{array}\right]\left[\begin{array}[]{rr}w_{\alpha}\\\ z_{\beta}\\\
\end{array}\right]=0.$ It follows that $\left[\begin{array}[]{rr}z_{\alpha}\\\
w_{\beta}\\\ \end{array}\right]=0.$ From 3.2, we get
$A^{\prime}_{\beta\alpha}z_{\alpha}+A^{\prime}_{\beta\beta}z_{\beta}=0.$ Hence
$A^{\prime}_{\beta\beta}z_{\beta}=0$ implies that $z_{\beta}=0$ as
$A^{\prime}_{\beta\beta}=A/A_{\alpha\alpha}$ is nonsingular. Clearly,
$w_{\alpha}=0.$ Hence $\left[\begin{array}[]{rr}w_{\alpha}\\\
w_{\beta}\end{array}\right]=\left[\begin{array}[]{rr}A^{\prime}_{\alpha\alpha}&A^{\prime}_{\alpha\beta}\\\
A^{\prime}_{\beta\alpha}&A^{\prime}_{\beta\beta}\\\
\end{array}\right]\left[\begin{array}[]{rr}z_{\alpha}\\\ z_{\beta}\\\
\end{array}\right]=0.$ Therefore $A^{\prime}$ is column competent. ∎

######  Theorem 3.10.

Let $A$ be a column competent matrix where $A_{\alpha\alpha}$ and the Schur
complement $A/A_{\alpha\alpha}$ be nonsingular of the square matrix
$A=\left[\begin{array}[]{rr}A_{\alpha\alpha}&A_{\alpha\beta}\\\
A_{\beta\alpha}&A_{\beta\beta}\\\ \end{array}\right].$ If $A\in E_{0}\cap
R_{0},$ then $A$ is column adequate.

###### Proof.

Suppose $A$ is not a column adequate matrix but is column competent. By
Theorem 2.2, $A$ is not a $P_{0}$\- matrix. Then there exists
$\beta\subseteq\\{1,2,\cdots,n\\}$ such that $\det A_{\beta\beta}<0.$ Let
$A\in E_{0}\cap R_{0}.$ It follows from the Theorem 2.3 that $A\in R.$ Then
deg${}_{A}(q)=1$ for any $q.$ Let $A^{\prime}$ be a principal pivot transform
of $A.$ Then $A^{\prime}\in R.$ Hence deg${}_{A^{\prime}}(q^{\prime})=1.$ By
Theorem 2.4, deg${}_{A^{\prime}}(q^{\prime})=$ deg${}_{A}(q).sgn(\det
A_{\beta\beta}).$ It implies that deg${}_{A^{\prime}}(q^{\prime})=-1.$ This
contradicts that $A$ is not a $P_{0}$-matrix. Therefore $A$ is column adequate
matrix. ∎

###  3.1 Solution of Linear Complementarity Problem with Column Competent
Matrices

We begin with some examples of $w$-uniqueness of the solution. Consider the
column competent matrix $A=\left[\begin{array}[]{rr}-1&3\\\ 2&-6\\\
\end{array}\right],q=\left[\begin{array}[]{rr}1\\\ -2\\\ \end{array}\right].$
This LCP$(q,A)$ has solution $z=\left[\begin{array}[]{rr}4\\\ 1\\\
\end{array}\right]$ and $w=\left[\begin{array}[]{rr}0\\\ 0\\\
\end{array}\right].$ In the neighbourhood of $z$ there is another solution
$z^{\prime}=\left[\begin{array}[]{rr}4.0100\\\ 1.0033\\\ \end{array}\right]$
and $w^{\prime}=w=\left[\begin{array}[]{rr}0\\\ 0\\\ \end{array}\right].$
We consider another matrix $A=\left[\begin{array}[]{rrr}-2&1&3\\\ 4&-2&-6\\\
1&-1&-1\\\ \end{array}\right],q=\left[\begin{array}[]{rrr}1\\\ -2\\\ 1\\\
\end{array}\right].$ For $z=\left[\begin{array}[]{rrr}2k\\\ k\\\ k\\\
\end{array}\right],$ $k\in R,z_{i}(Az)_{i}=0,\ i=1,2,\cdots,n$ implies that
$Az=0.$ So $A$ is a column competent matrix. This LCP$(q,A)$ has solution
$z=\left[\begin{array}[]{rrr}4\\\ 4\\\ 1\\\ \end{array}\right]$ and
$w=\left[\begin{array}[]{rr}0\\\ 0\\\ 0\\\ \end{array}\right].$ In the
neighbourhood of $z$ there is another solution
$z^{\prime}=\left[\begin{array}[]{rr}4.02\\\ 4.01\\\ 1.01\\\
\end{array}\right]$ and $w^{\prime}=w=\left[\begin{array}[]{rr}0\\\ 0\\\ 0\\\
\end{array}\right].$

Now we prove the following two results in connection with locally
$w$-uniqueness property of the column competent matrices. The following two
results state the necessary and sufficient condition that $A$ is a column
competent matrix in the system of linear complementarity problem.

######  Theorem 3.11.

Suppose $(w^{*},z^{*})$ is the solution of LCP$(q,A)$ such that
$w^{*}=q+Az^{*}.$ Let $\alpha=\\{i:{w_{i}}^{*}>0\\},$
$\beta=\\{i:{w_{i}}^{*}=0\\}$ be the index set. Further consider that the
submatrix $A_{\alpha\alpha}$ is nonsingular. If
$A=\left[\begin{array}[]{rr}A_{\alpha\alpha}&A_{\alpha\beta}\\\
A_{\beta\alpha}&A_{\beta\beta}\\\ \end{array}\right]$ is a column competent
matrix, then $(w_{\alpha},z_{\beta})=(0,0)$ is the only solution of the
system:

$\begin{split}z_{\alpha}=A^{\prime}_{\alpha\alpha}w_{\alpha}+A^{\prime}_{\alpha\beta}z_{\beta}=0\\\
w_{\beta}=A^{\prime}_{\beta\alpha}w_{\alpha}+A^{\prime}_{\beta\beta}z_{\beta}=0\\\
w_{\alpha}>0\\\ z_{\beta}>0,\end{split}$ (3.4)

where $A^{\prime}_{\alpha\alpha}=(A_{\alpha\alpha})^{-1},$
$A^{\prime}_{\alpha\beta}=-(A_{\alpha\alpha})^{-1}A_{\alpha\beta},$
$A^{\prime}_{\beta\alpha}=A_{\beta\alpha}(A_{\alpha\alpha})^{-1}$ and
$A^{\prime}_{\beta\beta}=A_{\beta\beta}-A_{\beta\alpha}(A_{\alpha\alpha})^{-1}A_{\alpha\beta}.$

###### Proof.

Let $A$ be a column competent matrix. Then by Theorem 2.1, it is locally
$w$-unique. Suppose $w^{*}$ is locally unique solution of LCP$(q,A)$ such that
$w^{*}=q+Az^{*}$ and the system (3.4) has a nonzero solution
$(\bar{w}_{\alpha},\bar{z}_{\beta}).$
Now $\left[\begin{array}[]{rr}\bar{z}_{\alpha}\\\ \bar{w}_{\beta}\\\
\end{array}\right]=\left[\begin{array}[]{rr}A^{\prime}_{\alpha\alpha}&A^{\prime}_{\alpha\beta}\\\
A^{\prime}_{\beta\alpha}&A^{\prime}_{\beta\beta}\\\
\end{array}\right]\left[\begin{array}[]{rr}\bar{w}_{\alpha}\\\
\bar{z}_{\beta}\\\ \end{array}\right]=0$ implies that
$\left[\begin{array}[]{rr}\bar{w}_{\alpha}\\\ \bar{w}_{\beta}\\\
\end{array}\right]=\left[\begin{array}[]{rr}A_{\alpha\alpha}&A_{\alpha\beta}\\\
A_{\beta\alpha}&A_{\beta\beta}\\\
\end{array}\right]\left[\begin{array}[]{rr}\bar{z}_{\alpha}\\\
\bar{z}_{\beta}\\\ \end{array}\right].$ Clearly, $\bar{w}=A\bar{z}$ and
$(w^{*})^{T}\bar{z}=0,$ $(\bar{w})^{T}z^{*}=0.$ Hence
$(w^{*}+\lambda\bar{w},z^{*}+\lambda\bar{z})$ solves LCP$(q,A)$ for all
$\lambda\geq 0.$ This contradicts the local uniqueness of $w^{*}.$ Therefore,
$(w_{\alpha},z_{\beta})=(0,0)$ is the only solution of the system (3.4). ∎

######  Theorem 3.12.

Suppose $(w^{*},z^{*})$ is the solution of LCP$(q,A)$ such that
$w^{*}=q+Az^{*}$ where $\alpha=\\{i:{w_{i}}^{*}>0\\}$ and
$\beta=\\{i:{w_{i}}^{*}=0\\}.$ Further suppose
$\left[\begin{array}[]{rr}{z}_{\alpha}\\\ {w}_{\beta}\\\
\end{array}\right]=\left[\begin{array}[]{rr}A^{\prime}_{\alpha\alpha}&A^{\prime}_{\alpha\beta}\\\
A^{\prime}_{\beta\alpha}&A^{\prime}_{\beta\beta}\\\
\end{array}\right]\left[\begin{array}[]{rr}{w}_{\alpha}\\\ {z}_{\beta}\\\
\end{array}\right]=0,$ $w_{\alpha}>0,z_{\beta}>0.$ If
$(z_{\alpha},z_{\beta})=(0,0)$ is the only solution of
$w_{\beta}=A_{\beta\alpha}z_{\alpha}+A_{\beta\beta}z_{\beta}=0$ then
$A=\left[\begin{array}[]{rr}A_{\alpha\alpha}&A_{\alpha\beta}\\\
A_{\beta\alpha}&A_{\beta\beta}\end{array}\right]$ is column competent.

###### Proof.

Suppose the matrix $A$ is not column competent. So $w^{*}$ is not locally
unique. Now $\left[\begin{array}[]{rr}{z}_{\alpha}\\\ {w}_{\beta}\\\
\end{array}\right]=\left[\begin{array}[]{rr}A^{\prime}_{\alpha\alpha}&A^{\prime}_{\alpha\beta}\\\
A^{\prime}_{\beta\alpha}&A^{\prime}_{\beta\beta}\\\
\end{array}\right]\left[\begin{array}[]{rr}{w}_{\alpha}\\\ {z}_{\beta}\\\
\end{array}\right]=0$ implies that $\left[\begin{array}[]{rr}{w}_{\alpha}\\\
{w}_{\beta}\\\
\end{array}\right]=\left[\begin{array}[]{rr}A_{\alpha\alpha}&A_{\alpha\beta}\\\
A_{\beta\alpha}&A_{\beta\beta}\\\
\end{array}\right]\left[\begin{array}[]{rr}{z}_{\alpha}\\\ {z}_{\beta}\\\
\end{array}\right]$ and $(w^{*})^{T}{z}=0,$ $({w})^{T}z^{*}=0.$ Hence
$(w^{*}+\lambda{w},z^{*}+\lambda{z})$ solves LCP$(q,A)$ for all $\lambda\geq
0.$ Hence $\exists$ a sequence of vectors $\\{\bar{w}^{k}\\}$ converging to
$w^{*}$ such that each
$(\bar{w}^{k},\bar{z}^{k})=(w^{*}+\lambda^{k}{w},z^{*}+\lambda^{k}{z})$ is a
solution of LCP$(q,A)$ with $\bar{w}^{k}=q+A\bar{z}^{k}.$ Since
$\bar{w}^{k}\to w^{*}$ and $\bar{z}^{k}\to z^{*},$ it follows that
$\bar{w}_{\alpha}^{k}>0,\bar{z}_{\beta}^{k}>0.$ By complementarity
$\bar{z}_{\alpha}^{k}=0,\bar{w}_{\beta}^{k}=0.$ Consider
$v^{k}=\bar{w}^{k}-w^{*}$ and $u^{k}=\bar{z}^{k}-z^{*}.$ The normalized
sequence $\\{v^{k}/\|{v^{k}}\|\\}$ is bounded and converges to $v^{*}\neq 0$
as $k\to\infty.$ Similarly, the normalized sequence $\\{u^{k}/\|{u^{k}}\|\\}$
is bounded and converges to $u^{*}\neq 0$ as $k\to\infty.$ Now for all large
$k,$ we have
$\bar{w}_{\beta}^{k}-w_{\beta}^{*}=\lambda^{k}w_{\beta}=0=A_{\beta\alpha}u_{\alpha}^{k}+A_{\beta\beta}u_{\beta}^{k}.$
Thus dividing by $\|u^{k}\|$ and $k\to\infty,$ we have
$A_{\beta\alpha}{u_{\alpha}}^{*}+A_{\beta\beta}{u_{\beta}}^{*}=0.$ Therefore,
$u^{*}=\left[\begin{array}[]{rr}{u_{\alpha}}^{*}\\\ {u_{\beta}}^{*}\\\
\end{array}\right]\neq 0$ is the nonzero solution of system
$w_{\beta}=A_{\beta\alpha}z_{\alpha}+A_{\beta\beta}z_{\beta}=0.$ It
contradicts that $(z_{\alpha},z_{\beta})=(0,0)$ is the only solution of the
system $w_{\beta}=A_{\beta\alpha}z_{\alpha}+A_{\beta\beta}z_{\beta}=0.$ Hence
$A$ is column competent. ∎

##  4 Conclusion

The complementary condition is an important issue in operations research. The
concept of matrix theoretic approach helps to develop many theory of linear
complementary problem. In this study we consider column competent matrix in
the context of local $w$-uniqueness property which is important both for the
theory as well as solution method of complementarity problrm. The results
based on w-uniqueness and column competent matrix class motivate future study
and application in matrix theory.

##  5 Acknowledgement

The author A. Dutta is thankful to the Department of Science and Technology,
Govt. of India, INSPIRE Fellowship Scheme for financial support.

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