$\displaystyle p(x,0)$ $\displaystyle=p_{\infty}(1+M_{\infty}g(x))$ (69)
$\displaystyle\rho(x,0)$
$\displaystyle=\rho_{\infty}\bigg{(}\frac{p}{p_{\infty}}\bigg{)}^{\frac{1}{\gamma}}$
$\displaystyle u(x,0)$
$\displaystyle=u_{\infty}+\frac{2(a-a_{\infty})}{\gamma-1}$

using the following conditions:

$p_{\infty}=1,\quad\rho_{\infty}=1,\quad M_{\infty}=0.01,\quad
g(x)=c_{0}sin(2\pi x/l),\quad c_{0}=0.1$

Figure 3(a) shows the non-dimensional gauge pressure distributions
$(p(x,t)-p_{\infty})/p_{\infty}$ after one period using 16 points per
wavelength (ppw) and a CFL number $\sigma=0.125$, requiring $128$ timesteps
for the acoustic and mixed fluxes, and $128/M=12,800$ timesteps for the
convective flux. The results for the acoustic and mixed schemes show the
expected behaviour: the wave has travelled almost one wavelength and has been
slightly diffused. The slight dispersion dispersion error of $2-3\%$ for the
acoustic and mixed schemes matches what would be expected from the result of
the von Neumann analysis in equations (62) and (66). The acoustic scheme has
diffused the pressure peaks roughly twice as much as the mixed scheme. The
convective scheme however has almost entirely smoothed out the wave, which has
travelled only a small portion of a wavelength. These results are entirely
consistent with our earlier analysis which found that the acoustic and mixed
schemes are both suitable for smooth acoustic flow, whereas the convective
scheme is overly diffusive of acoustic variations.

(a) Isolated soundwave after travelling for a single period.

(b) Low Mach number shocktube. Inset figure is a close-up of $16\leq x\leq
112$ and $0.96\leq y\leq 1.04$.

Figure 3: 1D acoustic examples

####  6.1.2 Low Mach number shocktube

The second one dimensional case is a low Mach number shock tube. This case
demonstrates the performance of the different fluxes for discontinuous purely
acoustic flow, and was first used by Sachdev et al. [43] for testing adaptive
schemes. The left and right initial states are:

$p_{L}=100028.04\text{Pa},\quad p_{R}=100000\text{Pa},\quad u_{L/R}=0,\quad
T_{L/R}=300\text{K}$

The very small pressure difference produces a contact wave moving to the right
at $M_{\infty}=0.0001$ between two receding weak shockwaves. Figure 3(b) shows
a close-up of the Mach number distributions between the two shock waves
obtained with the acoustic and the mixed flux schemes after $96$ timesteps
with a CFL of $\sigma=0.4$. The acoustic flux gives a monotone solution, as
expected for a first order, upwind scheme. However, the solution found with
the mixed flux has significant oscillations originating at the shockwaves - an
undamped grid mode as predicted by the von Neumann analysis.

From the one-dimensional examples we can verify that: the convective scheme is
completely unsuitable for flow with acoustic variations; the acoustic scheme
is suitable for flows with both smooth and discontinuous acoustic variations;
and the mixed scheme is suitable for flows with smooth acoustic variations,
but has too little diffusion to properly handle acoustic variations close to
the grid scale.

###  6.2 Two dimensional examples

Non-trivial solutions to the convective limit exist in two and higher
dimensions. We present two numerical examples in 2D, one which tests the
schemes’ capabilities for steady purely convective flow, and one which tests
the capabilities for unsteady mixed convective-acoustic flow.

####  6.2.1 Circular cylinder

This classic test case will demonstrate the performance of each flux scheme
for steady purely convective flow. The correct solutions should tend to the
incompressible solution as $M\to 0$. The farfield state is:

$\rho_{\infty}=1,\quad u_{\infty}=1,\quad M_{\infty}=0.01$

The background pressure is calculated from $p_{\infty}=\rho_{\infty}/(\gamma
M^{2}_{\infty})$. The cylinder is centred at the origin with radius $r=1$, and
is meshed using an O-mesh which extends out to 50 radii from the origin, with
64 and 48 cells in the circumferential and radial directions. The first row of
cells next to the cylinder has a radial height of $0.036r$. The cell height
grows at a rate of 1.11, with the final row having a radial height of $4.9r$.
Curvature-corrected boundary conditions are used at the inviscid cylinder wall
[74], and ghost cells at freestream boundaries are set using the upstream
velocity and entropy and the downstream static pressure. Convergence is
reached by pseudo-timestepping, which is accelerated by local timestepping and
preconditioning using Weiss & Smith’s preconditioning matrix [24]
(preconditioning is only used for the convective and mixed schemes, as
preconditioning does not reduce the spectral radius of the acoustic scheme
[20]).

The results for the convective flux is shown in figure 4(b), and closely
resemble the exact pressure distribution from the classical incompressible
potential solution in figure 4(a). The solution found with the acoustic flux
is shown in figure 4(c), and is visibly very different from the incompressible
solution. The artificial diffusion dominates over the other terms, and the
flow more closely resembles a Stokes flow than the inviscid solution, with
significantly larger pressure variations. Lastly, the solution found with the
mixed flux is shown in figure 4(d) and is very similar to the convective
scheme and the incompressible solution. However, on close inspection there is
a small chequer-board mode in the radial direction. This is more easily seen
in figure 5 which shows the difference between the exact pressure and the
pressure found with the mixed and convective flux schemes. The error for the
convective flux is smooth with no chequer-board modes, however the error for
the mixed flux clearly shows chequer-board modes in the radial direction.

(a) Exact solution

(b) Convective scheme

(c) Acoustic scheme

(d) Mixed scheme

Figure 4: Gauge pressure around a 2D cylinder at $M=0.01$ using the various
flux scalings.

(a) Convective scheme

(b) Mixed scheme

Figure 5: Gauge pressure error around a 2D cylinder for the convective and
mixed flux scalings.

####  6.2.2 Soundwave-vortex interaction

The last numerical example is the interaction of a one-dimensional soundwave
passing through a low Mach number Gresho vortex [75, 76] with stationary
background conditions. This provides a very simple test of a numerical
scheme’s ability to not ony simulate both convective and acoustic features,
but also their interaction, and has been used previously by [33, 54].

The domain is $(x,y)\in[-2h,2h)\times[-h,h)$ with periodic boundary conditions
in both dimensions, and is discretised with $256\times 128$ square cells. The
initial conditions are shown in figure 6(a), composed of the superposition of
a Gresho vortex centred at $(0,0)$ with a diameter $0.4h$, and a right-
travelling soundwave with a Gaussian profile centred along the line $(-h,y)$.
The soundwave profile is calculated according to equations (69), with:

$\rho_{\infty}=1,\quad M_{\infty}=0.01,\quad u_{\infty}=0,\quad
g(x)=c_{1}G(x+h)$

Where $G(x)$ is the Gaussian function with standard deviation of $0.1h$,
$c_{1}$ is chosen such that the maximum acoustic velocity is $1$, and
$p_{\infty}=\rho_{\infty}/(\gamma M^{2}_{\infty})$ again. The Gresho vortex
has circumferential velocity $u_{\theta}(r)$ and pressure $p(r)$ according to
[54]:

$u_{\theta}(r)=\begin{cases}5r,&0\leq r<0.2\\\ 2-5r,&0.2\leq r<0.4\\\
0,&0.4\leq r\end{cases}$ (70)
$p(r)=\begin{cases}p_{0}+\frac{25}{2}r^{2},&0\leq r<0.2\\\
p_{0}+\frac{25}{2}r^{2}+4(1-5r-\text{ln}0.2+\text{ln}r),&0.2\leq r<0.4\\\
p_{0}-2+4\text{ln}2,&0.4\leq r\end{cases}$ (71)

with $p_{0}=p_{\infty}+2-4\text{ln}2$.

The results for each flux after a single acoustic period ($4h/a_{\infty}$) and
after ten acoustic periods (equivalent to 1/10th of a convective period
$4h/(a_{\infty}M_{\infty})$) are shown in figures 6 and 7 respectively.

(a) Initial condition

(b) Convective scheme

(c) Acoustic scheme

(d) Mixed scheme

Figure 6: Velocity profiles for an acoustic wave and Gresho vortex after 1
acoustic period $\tau=l/a$ with various flux schemes.

Figure 6(b) shows the solution found with the convective scheme after one
acoustic time. The vortex shape is very well preserved, with barely any
reduction in peak velocity. The soundwave on the other hand has been
completely dissipated and is no longer visible. The only indication of the
soundwave is the slight asymmetry in the vortex velocity over the $y$ axis,
which happens as the soundwave is smeared out over the domain.

The solution from the acoustic scheme is shown in figure 6(c). The vortex has
a drastically reduced peak velocity, and is mishapen, having become aligned
with the grid due to the anisotropic artificial diffusion on the face-normal
velocity. The soundwave is still visible, having travelled once around the
domain. It is also damped, although not to an unsurprising degree for a first
order upwind scheme, given there were less than $20$ cells across the initial
width of the soundwave.

As predicted by the asymptotic analysis, the results for the mixed scheme
(figure 6(d)) combine the favourable behaviour of both the convective and
acoustic schemes. Both the vortex and the soundwave are well preserved. The
vortex shows comparable diffusion to the convective scheme (although without
the asymmetry), and the soundwave is less damped than that found with the
acoustic scheme, in agreement with the von Neumann analysis in the previous
section.

(a) Initial condition

(b) Convective scheme

(c) Acoustic scheme

(d) Mixed scheme

Figure 7: Velocity profiles for an acoustic wave and Gresho vortex after 10
acoustic periods (0.1 convective time periods $l/u$) with various flux
schemes.

The results for the convective scheme after 10 acoustic periods (0.1
convective periods) are shown in 7(b). The vortex shape is mostly preserved
with only minimal distortion around the corners, where the flow is most
misaligned with the grid. The peak vortex velocity is around $75\%$ of the
initial peak value. The acoustic scheme results are shown in figure 7(c). The
vortex is even more distorted than in figure 6(c), and has polluted a large
part of the domain. After integration over 10 periods, the soundwave is barely
visible on the left side of the figure behind this distortion. The result for
the mixed scheme again retains the favourable properties of the other two
schemes (figure 7(d)). The velocity magnitude of the vortex is similar to that
found with the convective scheme, but with marginally less distortion to the
shape. The soundwave is almost completely diffused, although less so than with
the acoustic scheme.

(a) Original mixed scheme

(b) Reduced upwinding

(c) Reduced $\nu_{22}$

(d) Reduced upwinding and $\nu_{22}$

Figure 8: Velocity profiles for an acoustic wave and Gresho vortex after 10
acoustic times $\tau$ with the mixed flux scheme, with various diffusion terms
reduced by a factor of $M$.

We now investigate the effect of reduced upwinding, as proposed in [29, 31,
32]. Figure 8(b) shows the solution after 10 acoustic times found with the
mixed scheme where the upwinding term in (40) has been reduced by a factor of
$M$. The peak vortex velocity is 87% of the initial value compared to 75% with
the original mixed scheme (figure 8(a)), a moderate improvement. Inspection of
the diffusion matrix (40) shows that for the mixed scheme, the $\nu_{22}$ term
is of the same magnitude as the upwind term, so contributes to the diffusion
of the convective components. Figure 8(c) shows the solution found by reducing
only the $\nu_{22}$ term by a factor of $M$. The peak velocity is 82% of the
initial value, a comparable improvement to reducing the upwind term, showing
that the $\nu_{22}$ term does indeed have a significant contribution to the
convective diffusion. Figure 8(d) shows the results found by reducing all
diffusion on the convective components (both the upwinding and $\nu_{22}$
terms) by a factor of $M$ compared to the original mixed scheme. The
improvement is striking, with a peak vortex velocity of 99% of the initial
value. Some small instabilities can be seem around the corners of the vortex,
where the flow is most misaligned with the grid and the 1D von Neumann
analysis of section 5 is least valid. Although figure 8(d) shows a significant
reduction in the diffusion without significant loss of stability, recall from
the von Neumann analysis that the stability of this scheme relies on the
timestep being proportional to the acoustic CFL, making it unsuitable for
preconditioned schemes where the timestep is proportional to the convective
CFL unless additional measures are taken to ensure stability.

The upwinding reductions in [31, 29, 32] do not exactly correspond to those
discussed here. Rieper [31] reduces the jump in the face-normal velocity
component by a factor of $M$, corresponding to reducing the $\nu_{22}$ term
from the acoustic scheme scaling to the mixed scheme scaling, and reducing by
a factor of $M$ the upwinding on the face-normal component of velocity only.
Thornber et al. [29] and Oßwald et al. [32] reduce the jumps in the all
velocity components by a factor of $M$, so additionally reduce the upwinding
of the face-transverse velocity components. For vortical flows these two
schemes are equivalent to the reduced upwinding scheme in figure 8(b), however
reducing the entire upwinding term also reduces the diffusion on the entropy
convection, so can be expected to give better results for cases with
significant entropy variations, for example simulations involving heat
transfer.

In this section we have used numerical examples to verify all of the findings
of the analyses in sections 3, 4 and 5. The one dimensional examples
demonstrated that the acoustic scheme is well suited to purely acoustic flow,
while the convective scheme is unable to resolve any acoustic phenomena. The
mixed scheme is suitable for smooth acoustic flow - resolving these phenomena
with less diffusion than the acoustic scheme - although is unstable for
discontinuous acoustic flow, with an undamped grid mode. Using steady flow
around a two dimensional cylinder, we verified that the acoustic scheme is
unsuitable for purely convective flow, whilst the convective scheme approaches
the incompressible limit as $M\to 0$. The mixed scheme also approaches the
incompressible limit, avoiding the catastrophic failure of the acoustic
scheme, however we saw it is susceptible to chequerboard modes on the second
order convective pressure. Finally we saw that, unsurprisingly, neither the
acoustic nor convective scheme is suitable for mixed convective-acoustic flow.
The mixed scheme on the other hand can resolve both convective and acoustic
features in the same flow, and can also achieve this with dramatically reduced
upwind diffusion, as shown in the von Neumann analysis of section 5.

##  7 Conclusions

Low Mach number flows include a range of applications of scientific and
engineering interest. Collocated, density based solvers for compressible flow
are one method of simulating low Mach number flow, but care must be taken to
ensure that the artificial diffusion does not compromise their accuracy in
this regime. In this paper we have reviewed the behaviour of the artificial
diffusion in this class of schemes at low Mach number using the modified
equations. By considering both the convective and acoustic low Mach number
limits, we have shown how three artificial diffusion scalings naturally arise
in the entropy variables - one suitable for purely convective flow, one for
purely acoustic flow, and one for flow with both convective and acoustic
features. Single- and multiple-scale asymptotic expansions of the modified
equations established the behaviours of each scheme for different flows. The
convective scaling is compatible with convective flows, but damps out acoustic
variations on a very fast timescale. The acoustic scaling is compatible with
acoustic flows, but will produce spurious pressure waves for convective flow.
The mixed scaling is compatible with both convective, acoustic, and mixed
flow, but has vanishing diffusion on the convective pressure and the acoustic
velocity, which may lead to pressure chequerboard modes and acoustic grid-mode
instabilities respectively.

Transforming the artificial diffusion to a Roe-type finite volume scheme in
the conserved variables allowed us to compare to a number of existing low Mach
number methods. Each of these methods matched one of the three scalings, and
our analysis agrees with previous theoretical analysis and well-known
behaviours of these existing methods. The convective and mixed scalings
conform with previous guidelines for accurate schemes for convective low Mach
number flow [30, 37], but by considering accuracy requirements for both
convective and acoustic effects, we were able to explain why there are two
possible scalings suitable for the convective limit, and what their relative
advantages and disadvantages are. Asymptotic expansion of the discrete
equations showed that all findings of the continuous analysis apply for the
first order Roe-type finite volume scheme, and von Neumann analysis confirmed
the stability estimates obtained from the continuous analysis. Finally, four
numerical examples demonstrated the performance of each diffusion scaling for
acoustic, convective and mixed convective-acoustic flow.

There is a significant body of literature investigating this class of schemes
in the limit of vanishing Mach number. We have shown that the most important
behaviours of this class of schemes at low Mach number can be found and
explained in a simple manner using the continuous modified equations in the
entropy variables. This form can be used to compare schemes and predict their
capabilities independently of the specific discretisation, as well as to
provide guidelines for the development of novel low-Mach schemes.

## Competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgements

The authors gratefully acknowledge support from Rolls-Royce plc and the EPSRC
Center for Doctoral Training in Gas Turbine Aerodynamics.

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##  Appendix A Transonic schemes at low Mach number

In this appendix we show, in a very approximate manner, why many numerical
schemes designed for transonic flow approach the acoustic diffusion scaling at
low Mach number. Diffusion schemes for the Euler equations can be classified
by how many waves they resolve. For example, scalar Lax-Friedrichs/Rusonov is
a 1-wave scheme, HLL is 2-wave scheme, and Roe, HLLC and AUSM are all 3-wave
schemes. In the characteristic variables, the diffusion Jacobian for the the
acoustic waves of a 2- or 3-waves scheme will be close to:

$\begin{matrix}\begin{pmatrix}|u+a|&0\\\ 0&|u-a|\\\ \end{pmatrix}\end{matrix}$
(72)

which at low Mach number can be approximated by:

$|u|\begin{matrix}\begin{pmatrix}M^{\scalebox{0.4}[1.0]{$-$}1}+1&0\\\
0&M^{\scalebox{0.4}[1.0]{$-$}1}-1\\\
\end{pmatrix}\end{matrix}\approx|u|\begin{matrix}\begin{pmatrix}M^{\scalebox{0.4}[1.0]{$-$}1}&0\\\
0&M^{\scalebox{0.4}[1.0]{$-$}1}\\\ \end{pmatrix}\end{matrix}$ (73)

The diffusion on the acoustic waves $du\pm dp/\rho a$ is then:

$(|u|M^{\scalebox{0.4}[1.0]{$-$}1})\underline{\underline{\mathcal{I}}}$ (74)

which matches the acoustic scaling. Because the diffusion (74) is a scalar
multiple of the identity matrix, any variable basis for the acoustic system
(for example $dp$ and $du$) will also have this diffusion at low Mach number.

##  Appendix B Non-dimensionalised artificial diffusion for finite-volume
schemes

The non-dimensional diffusion with convective scaling and
$\epsilon_{\alpha}=1$ is:

$\displaystyle f^{d}_{\rho}$
$\displaystyle=\frac{1}{2}\sum^{l}_{\mathcal{V}(i)}S_{il}\bigg{(}|U|\Delta_{il}\rho$
$\displaystyle+\rho\bigg{(}\frac{M^{\scalebox{0.4}[1.0]{$-$}2}\nu_{11}}{\rho|v|}\Delta_{il}p$
$\displaystyle+\nu_{12}\Delta_{il}U\bigg{)}\bigg{)}$ (75a) $\displaystyle
f^{d}_{\underline{\rho u}}$
$\displaystyle=\frac{1}{2}\sum^{l}_{\mathcal{V}(i)}S_{il}\bigg{(}|U|\Delta_{il}\underline{\rho
u}$ $\displaystyle+\underline{\rho
u}\bigg{(}\frac{M^{\scalebox{0.4}[1.0]{$-$}2}\nu_{11}}{\rho|v|}\Delta_{il}p$
$\displaystyle+\nu_{12}\Delta_{il}U\bigg{)}$
$\displaystyle+\underline{n}_{il}\big{(}M^{\scalebox{0.4}[1.0]{$-$}2}\nu_{21}\Delta_{il}p$
$\displaystyle+\rho|v|\nu_{22}\Delta_{il}U\big{)}\bigg{)}$ (75b)
$\displaystyle f^{d}_{\rho E}$
$\displaystyle=\frac{1}{2}\sum^{l}_{\mathcal{V}(i)}S_{il}\bigg{(}|U|\Delta_{il}\rho
E$ $\displaystyle+\rho
H\bigg{(}\frac{M^{\scalebox{0.4}[1.0]{$-$}2}\nu_{11}}{\rho|v|}\Delta_{il}p$
$\displaystyle+\nu_{12}\Delta_{il}U\bigg{)}$
$\displaystyle+UM^{2}\big{(}M^{\scalebox{0.4}[1.0]{$-$}2}\nu_{21}\Delta_{il}p$
$\displaystyle+\rho|v|\nu_{22}\Delta_{il}U\big{)}\bigg{)}$ (75c)

The non-dimensional diffusion with acoustic scaling and $\epsilon_{\alpha}=1$
is:

$\displaystyle f^{d}_{\rho}$
$\displaystyle=\frac{1}{2}\sum^{l}_{\mathcal{V}(i)}S_{il}\bigg{(}|U|\Delta_{il}\rho$
$\displaystyle+\rho\bigg{(}\frac{M^{\scalebox{0.4}[1.0]{$-$}1}\nu_{11}}{\rho|v|}\Delta_{il}p$
$\displaystyle+\nu_{12}\Delta_{il}U\bigg{)}\bigg{)}$ (76a) $\displaystyle
f^{d}_{\underline{\rho u}}$
$\displaystyle=\frac{1}{2}\sum^{l}_{\mathcal{V}(i)}S_{il}\bigg{(}|U|\Delta_{il}\underline{\rho
u}$ $\displaystyle+\underline{\rho
u}\bigg{(}\frac{M^{\scalebox{0.4}[1.0]{$-$}1}\nu_{11}}{\rho|v|}\Delta_{il}p$
$\displaystyle+\nu_{12}\Delta_{il}U\bigg{)}$
$\displaystyle+\underline{n}_{il}\big{(}M^{\scalebox{0.4}[1.0]{$-$}2}\nu_{21}\Delta_{il}p$
$\displaystyle+M^{\scalebox{0.4}[1.0]{$-$}1}\rho|v|\nu_{22}\Delta_{il}U\big{)}\bigg{)}$
(76b) $\displaystyle f^{d}_{\rho E}$
$\displaystyle=\frac{1}{2}\sum^{l}_{\mathcal{V}(i)}S_{il}\bigg{(}|U|\Delta_{il}\rho
E$ $\displaystyle+\rho
H\bigg{(}\frac{M^{\scalebox{0.4}[1.0]{$-$}1}\nu_{11}}{\rho|v|}\Delta_{il}p$
$\displaystyle+\nu_{12}\Delta_{il}U\bigg{)}$
$\displaystyle+UM^{2}\big{(}M^{\scalebox{0.4}[1.0]{$-$}2}\nu_{21}\Delta_{il}p$
$\displaystyle+M^{\scalebox{0.4}[1.0]{$-$}1}\rho|v|\nu_{22}\Delta_{il}U\big{)}\bigg{)}$
(76c)

The non-dimensional diffusion with blended scaling and $\epsilon_{\alpha}=1$
is:

$\displaystyle f^{d}_{\rho}$
$\displaystyle=\frac{1}{2}\sum^{l}_{\mathcal{V}(i)}S_{il}\bigg{(}|U|\Delta_{il}\rho$
$\displaystyle+\rho\bigg{(}\frac{M^{\scalebox{0.4}[1.0]{$-$}1}\nu_{11}}{\rho|v|}\Delta_{il}p$
$\displaystyle+\nu_{12}\Delta_{il}U\bigg{)}\bigg{)}$ (77a) $\displaystyle
f^{d}_{\underline{\rho u}}$
$\displaystyle=\frac{1}{2}\sum^{l}_{\mathcal{V}(i)}S_{il}\bigg{(}|U|\Delta_{il}\underline{\rho
u}$ $\displaystyle+\underline{\rho
u}\bigg{(}\frac{M^{\scalebox{0.4}[1.0]{$-$}1}\nu_{11}}{\rho|v|}\Delta_{il}p$
$\displaystyle+\nu_{12}\Delta_{il}U\bigg{)}$
$\displaystyle+\underline{n}_{il}\big{(}M^{\scalebox{0.4}[1.0]{$-$}2}\nu_{21}\Delta_{il}p$
$\displaystyle+\rho|v|\nu_{22}\Delta_{il}U\big{)}\bigg{)}$ (77b)
$\displaystyle f^{d}_{\rho E}$
$\displaystyle=\frac{1}{2}\sum^{l}_{\mathcal{V}(i)}S_{il}\bigg{(}|U|\Delta_{il}\rho
E$ $\displaystyle+\rho
H\bigg{(}\frac{M^{\scalebox{0.4}[1.0]{$-$}1}\nu_{11}}{\rho|v|}\Delta_{il}p$
$\displaystyle+\nu_{12}\Delta_{il}U\bigg{)}$
$\displaystyle+UM^{2}\big{(}M^{\scalebox{0.4}[1.0]{$-$}2}\nu_{21}\Delta_{il}p$
$\displaystyle+\rho|v|\nu_{22}\Delta_{il}U\big{)}\bigg{)}$ (77c)


# The Characteristic Polynomial of Projections

Kate Howell  Kate Howell: Department of Mathematics and Statistics, SUNY at
Albany, Albany, NY 12222, U.S.A<EMAIL_ADDRESS>and  Rongwei Yang
Rongwei Yang: Department of Mathematics and Statistics, SUNY at Albany,
Albany, NY 12222, U.S.A<EMAIL_ADDRESS>
###### Abstract.

This paper proves that the characteristic polynomial is a complete unitary
invariant for pairs of projection matrices. Some special cases involving three
or more projections are also considered.

##  0\. Introduction

The characteristic polynomial is an important topic in linear algebra and
matrix theory, and it has a wide range of applications in mathematics,
science, and engineering. A multivariable version of characteristic
polynomial, on the other hand, has been ostensibly missing until recently. The
challenge lies in the difficulty to define a computable notion when involved
matrices are noncommuting. Motivated by the early works on group
representation theory ([8, 7]), determinantal representations ([6, 12]), and
the projective joint spectrum of linear operators ([13]), a proper notion of a
multivariable characteristic polynomial seems to settle as follows.

######  Definition 0.1.

Given several $k\times k$ matrices $A_{1},...,A_{n}$, their characteristic
polynomial is defined as
$Q_{A}(z)=\det(z_{0}I+z_{1}A_{1}+\cdots+z_{n}A_{n}),z=(z_{0},...,z_{n})\in{\mathbb{C}}^{n+1}$.

Here, the subscript “$A$” in $Q_{A}$ refers to the linear pencil
$A(z):=z_{0}I+z_{1}A_{1}+\cdots+z_{n}A_{n}$. Recent studies have shown that
$Q_{A}$ offers an efficient tool for us to describe the interactions as well
as the joint behavior of the involved matrices. A few facts will substantiate
this claim. First, it is not hard to see that if $Q_{A}$ is irreducible, then
$A_{1},...,A_{n}$ have no nontrivial common invariant subspace. Further, if
$Q_{A}$ has a linear factor $z_{0}+\lambda_{1}z_{1}+...+\lambda_{n}z_{n}$,
then $\lambda_{j}$ is an eigenvalue of $A_{j}$, $1\leq j\leq n$. This becomes
apparent if one sets variables $z_{k}=0$ for all $k\neq 0,j$. Some deeper
results are as follows.

1. If $A_{1},...,A_{n}$ are normal matrices, then they commute if and only if $Q_{A}(z)$ is a product of linear factors ([5]).

2. If $A_{1},...,A_{n}$ are Coxeter generators of a finite Coxeter group, then $Q_{A}$ determines the group up to isomorphism ([4]).

3. If $G=\\{1,g_{1},...,g_{n}\\}$ is a finite group and $A_{i}=\lambda(g_{i})$, where $\lambda$ is the regular representation of $G$, then there is a one-to-one correspondence between the distinct factors of $Q_{A}$ and the equivalence classes of irreducible representations of $G$ ([8]).

4. If ${\mathcal{L}}$ is an $n$-dimensional complex Lie algebra and $A_{1},...,A_{n}$ are the adjoint representations of a basis, then $Q_{A}$ is invariant under the automorphism group of ${\mathcal{L}}$ ([1]). Moreover, ${\mathcal{L}}$ is solvable if and only if $Q_{A}(z)$ a product of linear factors ([11]).

5. Finite dimensional simple Lie algebras can be classified by their characteristic polynomials ([10]).

This paper aims to study the characteristic polynomial of several projection
matrices, and the main result is Theorem 2.3 which asserts that the
characteristic polynomial is a complete invariant for pairs of projection
matrices. Some relevant issues are also discussed.

Acknowledgement. The first author would like to thank her husband, Jason
Howell, for his support, encouragement, and prayers.

##  1\. Preliminaries

Given a $k\times k$ matrix $A$, we define $c_{0}(A)=1$ and

$c_{m}(A)=\frac{1}{m!}\det\begin{pmatrix}\operatorname{tr}A&m-1&0&\cdots&0\\\
\operatorname{tr}A^{2}&\operatorname{tr}A&m-2&\cdots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
\operatorname{tr}A^{m-1}&\operatorname{tr}A^{m-2}&\cdots&\cdots&1\\\
\operatorname{tr}A^{m}&\operatorname{tr}A^{m-1}&\cdots&\cdots&\operatorname{tr}A\end{pmatrix},\
\ 1\leq m\leq k.$ (1.1)

Then the Plemelj-Smithies formula ([9]), expresses the characteristic
polynomial of $A$ in the form

$\det(\lambda+A)=\sum_{m=0}^{k}\lambda^{k-m}c_{m}(A).$ (1.2)

Hence, for matrices $A_{1},\dots,A_{n}\in M_{k}({\mathbb{C}})$, if we set
$A_{*}(z^{\prime})=z_{1}A_{1}+\cdots+z_{n}A_{n}$, where
$z^{\prime}=(z_{1},...,z_{n})$, the function $Q_{A}(z_{0},z^{\prime})$ is the
classical characteristic polynomial of the matrix $A_{*}(z^{\prime})$. The
following fact is thus immediate.

######  Proposition 1.1.

Assume $A_{j},B_{j}\in M_{k}({\mathbb{C}}),1\leq j\leq n$, and $Q_{A}=Q_{B}$.
Then for each $z^{\prime}\in{\mathbb{C}}^{n}$ the matrices $A_{*}(z^{\prime})$
and $B_{*}(z^{\prime})$ have the same set of eigenvalues counting
multiplicity.

In particular, in this case the classical spectra
$\sigma(A_{j})=\sigma(B_{j})$ for each $j$. Expanding $Q_{A}$ as a polynomial
in $z_{0}$, one may write

$Q_{A}(z)=\sum_{m=0}^{k}z_{0}^{k-m}q_{m}(z^{{}^{\prime}}),\ \ \
z\in{\mathbb{C}}^{n+1}.$ (1.3)

where $q_{m}(z^{\prime})=c_{m}(A_{*}(z^{\prime}))$. The following lemma is
thus obtained by applying the Plemelj-Smithies formula to $A_{*}(z^{\prime})$.

######  Lemma 1.2.

For any $k\times k$ matrices $A_{1},...,A_{n}$, we have

  1. a)

$q_{1}(z^{\prime})=\sum^{n}_{j=1}z_{j}\textup{tr}A_{j}$.

  2. b)

$q_{2}(z^{\prime})=\frac{1}{2}\sum^{n}_{i,j=1}z_{i}z_{j}(\textup{tr}A_{i}\textup{tr}A_{j}-\textup{tr}(A_{i}A_{j}))$.

With greater effort, $q_{m}$ for $m\geq 3$ can also be determined. However, a
more elegant description of $q_{m}$ can be given by the cofactor matrix. Given
a general matrix $A=(a_{ij})\in M_{k}({\mathbb{C}})$, we let $C=(C_{ij})$ be
its cofactor matrix, e.g., $C_{ij}=(-1)^{i+j}\det(A_{ij})$, where $A_{ij}$ is
the $(i,j)$th minor of $A$. The cofactor theorem states that

$\det A=\displaystyle\sum_{m=1}^{k}a_{im}C_{im},\ \ \ 1\leq i\leq k.$ (1.4)

Treating $\det A$ as a multivariable function in its entries $a_{ij},1\leq
i,j\leq k$, we obtain

$\displaystyle\dfrac{\partial\det A}{\partial a_{ij}}$
$\displaystyle=\sum_{m=1}^{k}\bigg{(}\dfrac{\partial a_{im}}{\partial
a_{ij}}C_{im}+a_{im}\dfrac{\partial C_{im}}{\partial a_{ij}}\bigg{)}$
$\displaystyle=C_{ij}+\sum_{m=1}^{k}a_{im}\dfrac{\partial C_{im}}{\partial
a_{ij}}.$

Since $C_{im}$ is independent of the $i$th row of $A$, we have
$\dfrac{\partial C_{im}}{\partial a_{ij}}=0$ for every $m$. It follows that
$\dfrac{\partial\det A}{\partial a_{ij}}=C_{ij},1\leq i,j\leq k$. Thus one
obtains

$d\det(A)=\displaystyle\sum_{i,j=1}^{n}C_{ij}da_{ij}=\operatorname{tr}(C^{T}dA)$
(1.5)

which is well-known to experts. In the case $A$ is invertible, dividing (1.5)
by $\det A$, we have the Jacobi’s formula

$d\log\det A=\operatorname{tr}\left(A^{-1}dA\right).$ (1.6)

We now take another look at the polynomials $q_{1},...,q_{k}$ in the expansion
(1.3) of the characteristic polynomial $Q_{A}$. The cofactor matrix for the
linear pencil $A(z)$ is denoted by $C_{A}(z)$. Interestingly, its trace is the
generating function for $q_{1},...,q_{k}$ in the following sense.

######  Proposition 1.3.

For $1\leq m\leq k-1$, we have

$q_{k-m}(z^{\prime})=\frac{1}{m!}\Bigg{(}\frac{\partial^{m-1}}{\partial
z_{0}^{m-1}}\operatorname{tr}C_{A}(z)\Bigg{)}\Bigg{|}_{z_{0}=0}.$

###### Proof.

In (1.5), we substitute the general matrix $A$ by the linear pencil $A(z)$ to
obtain

$dQ_{A}(z)=\operatorname{tr}\left(C_{A}^{T}(z)(dz_{0}+A_{1}dz_{1}+\cdots+A_{n}dz_{n})\right).$

It follows that

$\frac{\partial Q_{A}(z)}{\partial
z_{0}}=\operatorname{tr}C_{A}^{T}(z)=\operatorname{tr}C_{A}(z).$ (1.7)

Setting $z_{0}=0$, we obtain
$q_{k-1}(z^{\prime})=\operatorname{tr}C_{A}(z)\big{|}_{z_{0}=0}$. In view of
the expansion (1.3), the formula for $q_{k-m}(z^{\prime})$ can be obtained by
taking the $(m-1)$th partial derivartive of (1.7) with respect to $z_{0}$ and
then setting $z_{0}=0$. ∎

##  2\. Two Projections

This section proves the main theorem of this paper. A complex square matrix
$p$ is a projection if $p^{*}=p$ and $p^{2}=p$. We begin by noting the
following definition.

######  Definition 2.1.

Two tuples of projections $(p_{1},...,p_{n})$ and $(q_{1},...,q_{n})$ are
unitarily equivalent if there exists a unitary matrix $U$ such that
$Up_{i}=q_{i}U$ for all $i$.

The following lemma from [3] is instrumental.

######  Lemma 2.2.

Two pairs of projection matrices $(p_{1},p_{2})$ and $(q_{1},q_{2})$ are
unitarily equivalent if and only if
$\operatorname{tr}(p_{1})=\operatorname{tr}(q_{1})$,
$\operatorname{tr}(p_{2})=\operatorname{tr}(q_{2})$, and
$\operatorname{tr}[(p_{1}p_{2})^{j}]=\operatorname{tr}[(q_{1}q_{2})^{j}]$ for
all $j\geq 1$.

Note that if the projections are of size $k\times k$, then due to the Cayley-
Hamilton theorem $(p_{1}p_{2})^{k}$ can be expressed as a linear combination
of lower degree terms. Thus, one only needs to check
$\operatorname{tr}[(p_{1}p_{2})^{j}]=\operatorname{tr}[(q_{1}q_{2})^{j}]$ for
each $j\leq k-1$. Pairs of projections have been well-studied in the
literature. For an extensive treatise on this subject, we refer the readers to
[2]. The main result of this paper is as follows.

######  Theorem 2.3.

Let $p=(p_{1},p_{2})$ and $q=(q_{1},q_{2})$ be two pairs of projection
matrices. Then they are unitarily equivalent if and only if $Q_{p}=Q_{q}$.

###### Proof.

The sufficiency is trivial. For the necessity, assume
$Q_{p}=\det(z_{0}I+z_{1}p_{1}+z_{2}p_{2})=\det(z_{0}I+z_{1}q_{1}+z_{2}q_{2})=Q_{q}$
and let $A:=z_{1}p_{1}+z_{2}p_{2}$ and $B:=z_{1}q_{1}+z_{2}q_{2}$. The first
two conditions of Lemma 2.2 follow readily from Lemma 1.2. In fact, since
$\operatorname{tr}(p_{1})z_{1}+\operatorname{tr}(p_{2})z_{2}=\operatorname{tr}(q_{1})z_{1}+\operatorname{tr}(q_{2})z_{2}$,
we have that $\operatorname{tr}(p_{1})=\operatorname{tr}(q_{1})$ and
$\operatorname{tr}(p_{2})=\operatorname{tr}(q_{2})$. To obtain the third
condition and thus the result of unitary equivalence, we will make three
claims. Each of these claims may be easily verified by induction, though the
notation is rather tedious for the first two and will therefore be excluded
from this paper.

######  Claim 1:

All terms in the fully expanded form of $A^{2n}$ reduce to the form
$p_{i}z_{1}^{s}z_{2}^{t}$, $p_{i}p_{j}p_{i}p_{j}\cdots
p_{i}p_{j}z_{1}^{s}z_{2}^{t}$, or $p_{i}p_{j}p_{i}p_{j}\cdots
p_{i}z_{1}^{s}z_{2}^{t}$, where $i,j\in\\{1,2\\}$, $i\neq j$,
$s,t\in\\{0,...,2n\\}$, and $s+t=2n$.

Taking the trace, we see that all coefficients in the fully expanded form of
$\operatorname{tr}(A^{2n})$ reduce to $\operatorname{tr}(p_{1})$,
$\operatorname{tr}(p_{2})$, or $\operatorname{tr}[(p_{1}p_{2})^{r}]$ for some
$r$.

######  Claim 2:

$(p_{1}p_{2})^{n}z_{1}^{n}z_{2}^{n}$ is the longest word that appears in the
fully expanded form of $A^{2n}$.

Taking the trace, we see $\operatorname{tr}[(p_{1}p_{2})^{n}]$ must appear in
$\operatorname{tr}(A^{2n})$ as a coefficient.

######  Claim 3:

$\operatorname{tr}[(p_{1}p_{2})^{j}]=\operatorname{tr}[(q_{1}q_{2})^{j}]$, for
all $j\geq 1.$

To prove claim 3, for the base case of $n=1$, consider
$\operatorname{tr}(A^{2})=\operatorname{tr}(B^{2})$. Then
$\operatorname{tr}(p_{1})z_{1}^{2}+2\operatorname{tr}(p_{1}p_{2})z_{1}z_{2}+\operatorname{tr}(p_{2})z_{2}^{2}=\operatorname{tr}(q_{1})z_{1}^{2}+2\operatorname{tr}(q_{1}q_{2})z_{1}z_{2}+\operatorname{tr}(q_{2})z_{2}^{2}$.
Hence, we have that
$\operatorname{tr}(p_{1}p_{2})=\operatorname{tr}(q_{1}q_{2})$. Now, assume the
hypothesis holds for $k\leq n$ and consider the fully expanded form of
$\operatorname{tr}(A^{2(n+1)})=\operatorname{tr}(B^{2(n+1)})$. From Claim 1,
we see that all coefficients must be of the form $\operatorname{tr}(p_{1})$,
$\operatorname{tr}(p_{2})$, or $\operatorname{tr}(p_{1}p_{2})^{r}$, and from
Claim 2 we see that $r\leq n+1$. Moreover, from Claim 2 we know that
$\operatorname{tr}(p_{1}p_{2})^{n+1}$ must appear as a coefficient at least
once. Hence, by factoring like $z^{s}z^{t}$ terms and noting that
$\operatorname{tr}(p_{1})=\operatorname{tr}(q_{1})$,
$\operatorname{tr}(p_{2})=\operatorname{tr}(q_{2})$, and
$\operatorname{tr}[(p_{1}p_{2})^{r}]=\operatorname{tr}[(q_{1}q_{2})^{r}]$,
$\forall r\leq n$ by assumption, we must have that
$\operatorname{tr}[(p_{1}p_{2})^{n+1}]=\operatorname{tr}[(q_{1}q_{2})^{n+1}]$.
This proves Claim 3, and the theorem follows from Lemma 2.2. ∎

With Theorem 2.3 at hand, we realize yet another property of the
characteristic polynomial: it is a complete invariant for pairs of
projections.

Now, let us consider the following application. Let $r=2p-I$, where $p$ is a
projection. A simple computation will show us that $r$ is an involution, i.e.,
$r^{2}=I$. On the other hand, given an involution $r$, $p=\frac{r+I}{2}$ is a
projection. Using this fact, we obtain the following corollary.

######  Corollary 2.4.

Two tuples of involutions $(r_{1},r_{2})$ and $(s_{1},s_{2})$ are unitarily
equivalent if their characteristic polynomials coincide.

##  3\. The Coxeter Group

We now move on to consider the characteristic polynomial of $n$ projections.
Interestingly, the Coxeter group plays an important role here. A finite
Coxeter group $W$ is a group generated by the set $S=\\{g_{1},...,g_{n}\\}$
which satisfies the relations $(g_{i}g_{j})^{m_{ij}}=1$ and $m_{ii}=1$, where
$m_{ij}\geq 2$ for $i\neq j$ and $i,j\in\\{1,...,n\\}$. The associated
$n\times n$ matrix $M=(m_{ij})$ is called the Coxeter matrix, and the pair
$(W,S)$ is called the Coxeter system. Notice that $M$ is symmetric since
$(g_{i}g_{j})^{-1}=g_{j}g_{i}$. It is possible for $m_{ij}$ to be infinite, in
which case the Coxeter group is infinite.

###  3.1. The Tits Representation

Given a Coxeter system $(W,S)$ and a basis $\\{e_{1},...,e_{n}\\}$ for
$\mathbb{C}^{n}$, the Tits representation (also called the reflection
representation) of $W$ is defined as

$\rho(g_{i})(e_{j})=e_{j}+2\cos\left(\frac{\pi}{m_{ij}}\right)e_{i},\ \ \
1\leq i,j\leq n.$

To more clearly see the matrix representation of $\rho(g_{i})$, we set
$\alpha_{ij}=\cos{\frac{\pi}{m_{ij}}}$. Observe that $\alpha_{ii}=-1$ and
$\alpha_{ij}\geq 0$ for $i\neq j$. Note in particular that if $m_{ij}$ is
infinite then $\alpha_{ij}=1$. With respect to the basis
$\\{e_{1},...,e_{n}\\}$, we have the matrix representation

$\rho(g_{i})=\begin{pmatrix}1&0&\ldots&0\\\ 0&1&\ldots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ \alpha_{i1}&\alpha_{i2}&\ldots&\alpha_{in}\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\ldots&1\end{pmatrix},\ \ \ i=1,...,n.$

Note that $\rho$ is not a unitary representation with respect to the ordinary
Euclidean metric on ${\mathbb{C}}^{n}$. However, it is with respect to the
inner product defined by $B(e_{i},e_{j})=-\cos(\frac{\pi}{m_{ij}})$ on
$\mathbb{C}^{n}$. Indeed, for all $1\leq i,j,k\leq n$, one verifies that

$\begin{split}&B(\rho(g_{k})e_{i},\rho(g_{k})e_{j})\\\
&=B\left(e_{i}+2\cos\left(\frac{\pi}{m_{ki}}\right)e_{k},e_{j}+2\cos\left(\frac{\pi}{m_{kj}}\right)e_{k}\right)\\\
&=B(e_{i},e_{j})+2\cos\left(\frac{\pi}{m_{kj}}\right)B(e_{i},e_{k})+2\cos\left(\frac{\pi}{m_{ki}}\right)B(e_{k},e_{j})\\\
&\hskip
58.32823pt+4\cos\left(\frac{\pi}{m_{ki}}\right)\cos\left(\frac{\pi}{m_{kj}}\right)B(e_{k},e_{k})\\\
&=B(e_{i},e_{j})-2\cos\left(\frac{\pi}{m_{kj}}\right)\cos\left(\frac{\pi}{m_{ki}}\right)-2\cos\left(\frac{\pi}{m_{ki}}\right)\cos\left(\frac{\pi}{m_{kj}}\right)\\\
&\hskip
58.32823pt+4\cos\left(\frac{\pi}{m_{ki}}\right)\cos\left(\frac{\pi}{m_{kj}}\right)=B(e_{i},e_{j}).\end{split}$

Recall from (1.3) that for the $n\times n$ matrices
$A_{i}=\rho(g_{i}),i=1,...,n$, the characteristic polynomial can be expressed
as $Q_{A}(z)=\sum_{m=0}^{n}z_{0}^{n-m}q_{m}(z^{\prime})$. Using Lemma 1.2, we
derive by direct computation the following general formulae:

$\displaystyle q_{1}(z^{\prime})$ $\displaystyle=(n-2)(z_{1}+\cdots+z_{n}),$
$\displaystyle q_{2}(z^{\prime})$
$\displaystyle=\frac{1}{2}(n-1)(n-4)(z_{1}^{2}+\cdots+z_{n}^{2})+\sum_{i<j}^{n}z_{i}z_{j}(n^{2}-5n+8-\alpha_{ij}^{2}).$
(3.1)

This leads to the following theorem.

######  Theorem 3.1.

A Coxeter system $(W,S)$ is uniquely determined, up to isomorphism, by its
characteristic polynomial with respect to the Tits representation.

###### Proof.

Let $(W,\\{g_{1},...,g_{n}\\})$ and
$(W^{\prime},\\{g^{\prime}_{1},...,g^{\prime}_{k}\\})$ be two Coxeter systems
with Tits representations $\rho$ and $\rho^{\prime}$, respectively. Set

$Q_{\rho}=\det(z_{0}I+z_{1}\rho(g_{1})+\cdots+\rho_{n}(g_{n}))$

and likewise for $Q_{\rho^{\prime}}$, and assume $Q_{\rho}=Q_{\rho^{\prime}}$.
It follows that

$k=\deg Q_{\rho^{\prime}}=\deg Q_{\rho}=n,\ \ \text{and}\ \
q_{2}=q^{\prime}_{2}.$

Since for each pair $i<j$ we have $(n^{2}-5n+8-\alpha_{ij}^{2})$ as the
coefficient of $z_{i}z_{j}$ in $q_{2}$, we must have
$\alpha_{ij}^{2}={\alpha^{\prime}}_{ij}^{2}$ and therefore
$\alpha_{ij}=\alpha^{\prime}_{ij}$. But since
$\alpha_{ij}=\cos{\frac{\pi}{m_{ij}}}$, we must have that
$m_{ij}=m^{\prime}_{ij}$ for all $i,j\in\\{1,...,n\\}$. Hence, by mapping
$g_{i}$ to $g^{\prime}_{i}$ for each $i$, we obtain an isomorphism between $W$
and $W^{\prime}$. ∎

###  3.2. Projections onto Hyperplanes

Let us now make a connection between the Tits representation and projections,
which will be essential in proving our next result. First, recall that the
Tits representation is unitary with respect to the inner product
$B(e_{i},e_{j})=-\cos(\frac{\pi}{m_{ij}})$ on $\mathbb{C}^{n}$. Moreover, for
each $i$ since $(\rho(g_{i}))^{2}=\rho(g_{i}^{2})=I$, we have that
$\rho(g_{i})=\rho^{-1}(g_{i})=\rho^{*}(g_{i})$, i.e, $\rho(g_{i})$ is an
involution. We set $p_{i}=\frac{I+\rho(g_{i})}{2}$. It is easy to check that
$p_{i}$ is the orthogonal projection onto the hyperplane
$\\{v\in{\mathbb{C}}^{n}\mid B(e_{i},v)=0\\}$.

######  Theorem 3.2.

Suppose $p=(p_{1},...,p_{n})$ and $q=(q_{1},...,q_{n})$ are tuples of linearly
independent projections onto hyperplanes in $\mathbb{C}^{n}$. Then $p$ and $q$
are unitarily equivalent if and only their characteristic polynomials
coincide.

###### Proof.

The necessity is trivial. To begin, we assume
$Q_{p}(z)=\det(z_{0}I+\cdots+z_{n}p_{n})$ and likewise for $Q_{q}$. For each
$1\leq i\leq n$, define $g_{i}=2p_{i}-I$ and $g^{\prime}_{i}=2q_{i}-I$. It is
easy to check that $g_{i}$ and $g_{i}^{\prime}$ are reflection matrices about
the hyperplanes $p_{i}({\mathbb{C}}^{n})$ and $q_{i}({\mathbb{C}}^{n})$,
respectively. Let $W=\langle g_{1},...,g_{n}\rangle$ and $W^{\prime}=\langle
g^{\prime}_{1},...,g^{\prime}_{n}\rangle$ be the Coxeter groups with
corresponding Coxeter matrices $M=(m_{ij})$ and resp.
$M^{\prime}=(m^{\prime}_{ij})$. Moreover, let $e_{i}$ (resp. $e_{i}^{\prime})$
be an unit normal vector to $p_{i}({\mathbb{C}}^{n})$ (resp.
$q_{i}({\mathbb{C}}^{n})$). Then $\\{e_{1},...,e_{n}\\}$ (resp.
$\\{e^{\prime}_{1},...,e^{\prime}_{n}\\}$) is linearly independent and hence a
basis of ${\mathbb{C}}^{n}$. Observe that the identity map is the Tits
representation for both $W$ and $W^{\prime}$ in this case. Suppose
$Q_{p}=Q_{q}$. Then $M=M^{\prime}$ by the proof of Theorem 3.1. Furthermore,
we have an isomorphism $\phi:W\rightarrow W^{\prime}$ defined by
$\phi(g_{i})=g^{\prime}_{i}$. Our goal from here will be to construct a
unitary $U$ and use it to explicitly show that $U^{*}g^{\prime}_{i}U=g_{i}$,
$\forall i\in\\{1,...,n\\}$, which would therefore imply the unitary
equivalence of $(p_{1},...,p_{n})$ and $(q_{1},...,q_{n})$. To proceed, we
define $U(e_{i})=e_{i}^{\prime},1\leq i\leq n$. Note that $U$ is unitary since

$B(e_{i},e_{j})=-\cos(\frac{\pi}{m_{ij}})=-\cos(\frac{\pi}{m^{\prime}_{ij}})=B(e^{\prime}_{i},e^{\prime}_{j}).$

Indeed, we see that $g_{i}$ and $g^{\prime}_{i}$ are unitarily equivalent
since

$\displaystyle U^{*}g^{\prime}_{i}U(e_{j})$
$\displaystyle=U^{*}\left(U(e_{j})+2\cos\left(\frac{\pi}{m^{\prime}_{ij}}\right)U(e_{i})\right)$
$\displaystyle=e_{j}+2\cos\left(\frac{\pi}{m_{ij}}\right)e_{i}=g_{i}(e_{j}).$

∎

The following is an immediate consequence.

######  Corollary 3.3.

Suppose $p=(p_{1},...,p_{n})$ and $q=(q_{1},...,q_{n})$ are tuples of linearly
independent rank-$1$ projections in $\mathbb{C}^{n}$. Then they are unitarily
equivalent if and only if their characteristic polynomials coincide.

In this case, since $I-p_{i}$ and $I-q_{i}$ are projections onto hyperplances,
the corollary follows directly from Theorem 3.2 after a change of variables in
the characteristic polynomial.

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