#  Projecting onto rectangular matrices with prescribed row and column sums

Heinz H. Bauschke, Shambhavi Singh, and Xianfu Wang  Mathematics, University
of British Columbia, Kelowna, B.C. V1V 1V7, Canada. E-mail:
<EMAIL_ADDRESS>Mathematics, University of British Columbia, Kelowna,
B.C. V1V 1V7, Canada. E-mail<EMAIL_ADDRESS>Mathematics, University
of British Columbia, Kelowna, B.C. V1V 1V7, Canada. E-mail<EMAIL_ADDRESS>
(October 15, 2021)

###### Abstract

In 1990, Romero presented a beautiful formula for the projection onto the set
of rectangular matrices with prescribed row and column sums. Variants of
Romero’s formula have been rediscovered by Khoury and by Glunt, Hayden, and
Reams, for bistochastic (square) matrices in 1998. These results have found
various generalizations and applications.

In this paper, we provide a formula for the more general problem of finding
the projection onto the set of rectangular matrices with prescribed _scaled_
row and column sums. Our approach is based on computing the Moore-Penrose
inverse of a certain linear operator associated with the problem. In fact, our
analysis holds even for Hilbert-Schmidt operators and we do not have to assume
consistency. We also perform numerical experiments featuring the new
projection operator.

2020 Mathematics Subject Classification: Primary 90C25; Secondary 15A10,
15B51, 47L07.

Keywords: generalized bistochastic matrices, Hilbert-Schmidt operators, Moore-
Penrose inverse, projection, rectangular matrices, transportation polytope.

##  1 Motivation

A matrix in $\mathbb{R}^{n\times n}$ is called _bistochastic_ if all entries
of it are nonnegative and all its row and column sums equal $1$. More
generally, a matrix is _generalized bistochastic_ if the requirement on
nonnegativity is dropped. The bistochastic matrices form a convex polytope
$B$, commonly called the _Birkhoff polytope_ , in $\mathbb{R}^{n\times n}$,
with its extreme points being the permutation matrices (a seminal result due
to Birkhoff and von Neumann). A lovely formula provided in 1998 by Khoury [8]
— and also by Glunt et al. [5] — gives the projection of any matrix onto $G$,
the affine subspace of generalized bistochastic matrices (see Example 3.8
below). More generally, nonnegative _rectangular_ matrices with prescribed row
and column sums are called _transportation polytopes_. If the nonnegativity
assumption is dropped, then Romero provided already in 1990 an explicit
formula (see Remark 3.5 below) which even predates the square case! On the
other hand, the projection onto the set of nonnegative matrices $N$ is simple
— just replace every negative entry by $0$. No explicit formula is known to
project a matrix onto the set of bistochastic matrices; however, because
$B=G\cap N$, one may apply algorithms such as Dykstra’s algorithm to
iteratively approximate the projection onto $B$ by using the projection
operators $P_{G}$ and $P_{N}$ (see, e.g., Takouda’s [12] for details). In the
case of transportation polytopes, algorithms which even converge in _finitely
many steps_ were provided by Calvillo and Romero [4].

The goal of this paper is to provide explicit projection operators in more
general settings. Specifically, we present a projection formula for finding
the projection onto the set of rectangular matrices with prescribed _scaled_
row and column sums. Such problems arise, e.g., in _discrete tomography_ [13]
and the study of _transportation polytopes_ [4]. Our approach uses the Moore-
Penrose inverse of a certain linear operator $\mathcal{A}$. It turns out that
our analysis works even for Hilbert-Schmidt operators because the range of
$\mathcal{A}$ can be determined and seen to be closed. Our main references are
[3], [7] (for Hilbert-Schmidt operators), and [6] (for the Moore-Penrose
inverse). We also note that consistency is not required.

The paper is organized as follows. After recording a useful result involving
the Moore-Penrose inverse at the end of this section, we prove our main
results in Section 2. These result are then specialized to rectangular
matrices in Section 3. We then turn to numerical experiments in Section 4
where we compare the performance of three popular algorithms: Douglas-
Rachford, Method of Alternating Projections, and Dykstra.

We conclude this introductory section with a result which we believe to be
part of the folklore (although we were not able to pinpoint a crisp
reference). It is formulated using the _Moore-Penrose inverse_ of an operator
— for the definition of the Moore-Penrose inverse and its basic properties,
see [6] (and also [3, pages 57–59] for a crash course). The formula presented
works even in the case when the problem is inconsistent and automatically
provides a least squares solution.

######  Proposition 1.1.

Let $A\colon X\to Y$ be a continuous linear operator with closed range between
two real Hilbert spaces. Let $b\in Y$, set
$\bar{b}:=P_{{\operatorname{ran}}\,A}b$, and set $C:=A^{-1}\bar{b}$. Then

$(\forall x\in X)\quad P_{C}x=x-A^{\dagger}(Ax-b),$ (1)

where $A^{\dagger}$ denotes the Moore-Penrose inverse of $A$.

Proof. Clearly, $\bar{b}\in{\operatorname{ran}}\,A$; hence,
$C\neq\varnothing$. Let $x\in X$. It is well known (see, e.g., [3, Example
29.17(ii)]) that

$P_{C}x=x-A^{\dagger}(Ax-\bar{b}).$ (2)

On the other hand,

$A^{\dagger}\bar{b}=A^{\dagger}P_{{\operatorname{ran}}\,A}b=A^{\dagger}AA^{\dagger}b=A^{\dagger}b$
(3)

using the fact that $AA^{\dagger}=P_{{\operatorname{ran}}\,A}$ (see, e.g., [3,
Proposition 3.30(ii)]) and $A^{\dagger}AA^{\dagger}=A^{\dagger}$ (see, e.g.,
[6, Section II.2]). Altogether, $P_{C}x=x-A^{\dagger}(Ax-b)$ as claimed.
$\hfill\quad\blacksquare$

##  2 Hilbert-Schmidt operators

From now on, we assume that

$\text{ $X$ and $Y$ are two real Hilbert spaces},$ (4)

which in turn give rise to the real Hilbert space

$\mathcal{H}:=\big{\\{}{T\colon X\to Y}~{}\big{|}~{}{\text{$T$ is Hilbert-
Schmidt}}\big{\\}}.$ (5)

Hilbert-Schmidt operators encompass rectangular matrices — even with
infinitely many entries as long as these are square summable — as well as
certain integral operators. (We refer the reader to [7, Section 2.6] for basic
results on Hilbert-Schmidt operators and also recommend [10, Section VI.6].)
Moreover, $\mathcal{H}$ (is generated by and) contains rank-one operators of
the form

$(v\otimes u)\colon X\to Y\colon x\mapsto\left\langle{u},{x}\right\rangle v,$
(6)

where $(v,u)\in Y\times X$, and with adjoint

$(v\otimes u)^{*}\colon Y\to X\colon y\mapsto\left\langle{v},{y}\right\rangle
u$ (7)

and

$(v\otimes u)^{*}=u\otimes v.$ (8)

Moreover,

$u\otimes u=\|u\|^{2}P_{\,\mathbb{R}u}\quad\text{and}\quad v\otimes
v=\|v\|^{2}P_{\,\mathbb{R}v}.$ (9)

For the rest of the paper, we fix

$e\in X$ and $f\in Y$, (10)

and set

$\mathcal{A}\colon\mathcal{H}\to Y\times X\colon T\mapsto(Te,T^{*}f).$ (11)

######  Proposition 2.1.

$\mathcal{A}$ is a continuous linear operator and
$\displaystyle\|\mathcal{A}\|=\sqrt{\|e\|^{2}+\|f\|^{2}}$.

Proof. Clearly, $\mathcal{A}$ is a linear operator. Moreover, $(\forall
T\in\mathcal{H})$
$\|\mathcal{A}(T)\|^{2}=\|Te\|^{2}+\|T^{*}f\|^{2}\leq\|T\|_{\textsf{op}}^{2}\|e\|^{2}+\|T^{*}\|_{\textsf{op}}^{2}\|f\|^{2}\leq\|T\|^{2}(\|e\|^{2}+\|f\|^{2})$
because the Hilbert-Schmidt norm dominates the operator norm. It follows that
$\mathcal{A}$ is continuous and
$\|\mathcal{A}\|\leq\sqrt{\|e\|^{2}+\|f\|^{2}}$. On the other hand, if
$T=f\otimes e$, then $\|T\|=\|e\|\|f\|$,
$\mathcal{A}(T)=(\|e\|^{2}f,\|f\|^{2}e)$ and hence
$\|\mathcal{A}(T)\|=\|T\|\sqrt{\|e\|^{2}+\|f\|^{2}}$. Thus
$\|\mathcal{A}\|\geq\sqrt{\|e\|^{2}+\|f\|^{2}}$. Combining these observations,
we obtain altogether that $\|\mathcal{A}\|=\sqrt{\|e\|^{2}+\|f\|^{2}}$.
$\hfill\quad\blacksquare$

We now prove that ${\operatorname{ran}}\,\mathcal{A}$ is always closed.

######  Proposition 2.2.

(range of $\mathcal{A}$ is closed) The following hold:

  1. (i)

If $e=0$ and $f=0$, then
${\operatorname{ran}}\,\mathcal{A}=\\{0\\}\times\\{0\\}$.

  2. (ii)

If $e=0$ and $f\neq 0$, then ${\operatorname{ran}}\,\mathcal{A}=\\{0\\}\times
X$.

  3. (iii)

If $e\neq 0$ and $f=0$, then
${\operatorname{ran}}\,\mathcal{A}=Y\times\\{0\\}$.

  4. (iv)

If $e\neq 0$ and $f\neq 0$, then
${\operatorname{ran}}\,\mathcal{A}=\\{(f,-e)\\}^{\perp}$.

Consequently, ${\operatorname{ran}}\,\mathcal{A}$ is always a closed linear
subspace of $Y\times X$.

Proof. (i): Clear.

(ii): Obviously, ${\operatorname{ran}}\,\mathcal{A}\subseteq\\{0\\}\times X$.
Conversely, let $x\in X$ and set

$T:=\frac{1}{\|f\|^{2}}f\otimes x.$ (12)

Then $Te=T0=0$ and

$T^{*}f=\frac{1}{\|f\|^{2}}(f\otimes
x)^{*}f=\frac{1}{\|f\|^{2}}\left\langle{f},{f}\right\rangle x=x$ (13)

and thus
$(0,x)=(Te,T^{*}f)=\mathcal{A}(T)\in{\operatorname{ran}}\,\mathcal{A}$.

(iii): Obviously, ${\operatorname{ran}}\,\mathcal{A}\subseteq Y\times\\{0\\}$.
Conversely, let $y\in Y$ and set

$T:=\frac{1}{\|e\|^{2}}y\otimes e.$ (14)

Then $T^{*}f=T^{*}0=0$ and

$Te=\frac{1}{\|e\|^{2}}(y\otimes
e)e=\frac{1}{\|e\|^{2}}\left\langle{e},{e}\right\rangle y=y$ (15)

and thus
$(y,0)=(Te,T^{*}f)=\mathcal{A}(T)\in{\operatorname{ran}}\,\mathcal{A}$.

(iv): If $(y,x)\in{\operatorname{ran}}\,\mathcal{A}$, say
$(y,x)=\mathcal{A}(T)=(Te,T^{*}f)$ for some $T\in\mathcal{H}$, then

$\displaystyle\left\langle{f},{y}\right\rangle$
$\displaystyle=\left\langle{f},{Te}\right\rangle=\left\langle{T^{*}f},{e}\right\rangle=\left\langle{x},{e}\right\rangle,$
(16)

i.e., $(y,x)\perp(f,-e)$. It follows that
${\operatorname{ran}}\,\mathcal{A}\subseteq\\{(f,-e)\\}^{\perp}$.

Conversely, let $(y,x)\in\\{(f,-e)\\}^{\perp}$, i.e.,
$\left\langle{e},{x}\right\rangle=\left\langle{f},{y}\right\rangle$.

Case 1: $\left\langle{e},{x}\right\rangle=\left\langle{f},{y}\right\rangle\neq
0$.
Set

$\zeta:=\frac{1}{\left\langle{x},{e}\right\rangle}=\frac{1}{\left\langle{y},{f}\right\rangle}\quad\text{and}\quad
T:=\zeta(y\otimes x)\in\mathcal{H}.$ (17)

Note that

$Te=\zeta(y\otimes x)e=\zeta\left\langle{x},{e}\right\rangle y=y$ (18)

and

$T^{*}f=\zeta(y\otimes x)^{*}f=\zeta\left\langle{y},{f}\right\rangle x=x;$
(19)

therefore,
$(y,x)=(Te,T^{*}f)=\mathcal{A}(T)\in{\operatorname{ran}}\,\mathcal{A}$.

Case 2: $\left\langle{e},{x}\right\rangle=\left\langle{f},{y}\right\rangle=0$.
Pick $\xi$ and $\eta$ in $\mathbb{R}$ such that

$\xi\|f\|^{2}=1\quad\text{and}\quad\eta\|e\|^{2}=1,$ (20)

and set

$T:=\xi(f\otimes x)+\eta(y\otimes e)\in\mathcal{H}.$ (21)

Then

$Te=\xi(f\otimes x)e+\eta(y\otimes e)e=\xi\left\langle{x},{e}\right\rangle
f+\eta\left\langle{e},{e}\right\rangle y=0f+\eta\|e\|^{2}y=y$ (22)

and

$T^{*}f=\xi(f\otimes x)^{*}f+\eta(y\otimes
e)^{*}f=\xi\left\langle{f},{f}\right\rangle
x+\eta\left\langle{y},{f}\right\rangle e=\xi\|f\|^{2}x+0e=x.$ (23)

Thus $(y,x)=(Te,T^{*}f)=\mathcal{A}(T)\in{\operatorname{ran}}\,\mathcal{A}$.
$\hfill\quad\blacksquare$

We now turn to the adjoint of $\mathcal{A}$:

######  Proposition 2.3.

(adjoint of $\mathcal{A}$) We have

$\mathcal{A}^{*}\colon{Y}\times{X}\to\mathcal{H}\colon(y,x)\mapsto y\otimes
e+f\otimes x.$ (24)

Proof. Let $T\in\mathcal{H}$ and $(y,x)\in Y\times X$. Let $B$ be any
orthonormal basis of $X$. Then

$\displaystyle\left\langle{\mathcal{A}(T)},{(y,x)}\right\rangle$
$\displaystyle=\left\langle{(Te,T^{*}f)},{(y,x)}\right\rangle$ (25a)
$\displaystyle=\left\langle{Te},{y}\right\rangle+\left\langle{T^{*}f},{x}\right\rangle$
(25b)
$\displaystyle=\left\langle{e},{T^{*}y}\right\rangle+\left\langle{T^{*}f},{x}\right\rangle$
(25c) $\displaystyle=\sum_{b\in
B}\big{(}\left\langle{e},{b}\right\rangle\left\langle{b},{T^{*}y}\right\rangle+\left\langle{T^{*}f},{b}\right\rangle\left\langle{b},{x}\right\rangle\big{)}$
(25d) $\displaystyle=\sum_{b\in
B}\left\langle{Tb},{\left\langle{e},{b}\right\rangle
y}\right\rangle+\sum_{b\in
B}\left\langle{Tb},{\left\langle{x},{b}\right\rangle f}\right\rangle$ (25e)
$\displaystyle=\sum_{b\in B}\left\langle{Tb},{(y\otimes
e)b}\right\rangle+\sum_{b\in B}\left\langle{Tb},{(f\otimes x)b}\right\rangle$
(25f) $\displaystyle=\left\langle{T},{y\otimes
e}\right\rangle+\left\langle{T},{f\otimes x}\right\rangle$ (25g)
$\displaystyle=\left\langle{T},{y\otimes e+f\otimes x}\right\rangle$ (25h)

which proves the result. $\hfill\quad\blacksquare$

We have all the results together to start tackling the Moore-Penrose inverse
of $\mathcal{A}$.

######  Theorem 2.4.

(Moore-Penrose inverse of $\mathcal{A}$ part 1) Suppose that $e\neq 0$ and
$f\neq 0$. Let $(y,x)\in Y\times X$. Then

$\mathcal{A}^{\dagger}(y,x)=\frac{1}{\|e\|^{2}}\Big{(}y\otimes
e-\frac{\left\langle{f},{{y}}\right\rangle}{\|e\|^{2}+\|f\|^{2}}f\otimes
e\Big{)}+\frac{1}{\|f\|^{2}}\Big{(}f\otimes
x-\frac{\left\langle{e},{x}\right\rangle}{\|e\|^{2}+\|f\|^{2}}f\otimes
e\Big{)}.$ (26)

Proof. Set

$(v,u):=\bigg{(}\frac{1}{\|e\|^{2}}\Big{(}y-\frac{\left\langle{y},{f}\right\rangle}{\|e\|^{2}+\|f\|^{2}}f\Big{)},\frac{1}{\|f\|^{2}}\Big{(}x-\frac{\left\langle{x},{e}\right\rangle}{\|e\|^{2}+\|f\|^{2}}e\Big{)}\bigg{)}.$
(27)

Then

$\displaystyle\left\langle{f},{v}\right\rangle$
$\displaystyle=\frac{1}{\|e\|^{2}}\Big{(}\left\langle{f},{y}\right\rangle-\frac{\left\langle{y},{f}\right\rangle}{\|e\|^{2}+\|f\|^{2}}\left\langle{f},{f}\right\rangle\Big{)}$
(28a)
$\displaystyle=\frac{\left\langle{f},{y}\right\rangle}{\|e\|^{2}}\Big{(}1-\frac{\|f\|^{2}}{\|e\|^{2}+\|f\|^{2}}\Big{)}$
(28b)
$\displaystyle=\frac{\left\langle{f},{y}\right\rangle}{\|e\|^{2}}\cdot\frac{\|e\|^{2}+\|f\|^{2}-\|f\|^{2}}{\|e\|^{2}+\|f\|^{2}}$
(28c)
$\displaystyle=\frac{\left\langle{f},{y}\right\rangle}{\|e\|^{2}+\|f\|^{2}},$
(28d)

and similarly

$\left\langle{e},{u}\right\rangle=\frac{\left\langle{e},{x}\right\rangle}{\|e\|^{2}+\|f\|^{2}}.$
(29)

Substituting 28 and 29 in 27 yields

$(v,u)=\Big{(}\frac{1}{\|e\|^{2}}\big{(}y-\left\langle{f},{{v}}\right\rangle
f\big{)},\frac{1}{\|f\|^{2}}\big{(}x-\left\langle{e},{u}\right\rangle
e\big{)}\Big{)}.$ (30)

Thus

$y=\|e\|^{2}v+\left\langle{f},{v}\right\rangle f\quad\text{and}\quad
x=\|f\|^{2}u+\left\langle{e},{u}\right\rangle e.$ (31)

Therefore, using 24, 30, 7, and 24 again, we obtain

$\displaystyle\mathcal{A}^{*}\mathcal{A}\mathcal{A}^{*}(v,u)$
$\displaystyle=\mathcal{A}^{*}\mathcal{A}(v\otimes e+f\otimes u)$ (32a)
$\displaystyle=\mathcal{A}^{*}\big{(}(v\otimes e)e+(f\otimes u)e,(v\otimes
e)^{*}f+(f\otimes u)^{*}f\big{)}$ (32b)
$\displaystyle=\mathcal{A}^{*}\big{(}\|e\|^{2}v+\left\langle{e},{u}\right\rangle
f,\left\langle{f},{v}\right\rangle e+\|f\|^{2}u\big{)}$ (32c)
$\displaystyle=\big{(}\|e\|^{2}v+\left\langle{e},{u}\right\rangle
f\big{)}\otimes e+f\otimes\big{(}\left\langle{f},{v}\right\rangle
e+\|f\|^{2}u\big{)}$ (32d) $\displaystyle=\|e\|^{2}v\otimes
e+\left\langle{e},{u}\right\rangle f\otimes e+\left\langle{f},{v}\right\rangle
f\otimes e+\|f\|^{2}f\otimes u$ (32e)
$\displaystyle=\big{(}\|e\|^{2}v+\left\langle{f},{v}\right\rangle
f\big{)}\otimes e+f\otimes\big{(}\|f\|^{2}u+\left\langle{e},{u}\right\rangle
e\big{)}$ (32f) $\displaystyle=y\otimes e+f\otimes x$ (32g)
$\displaystyle=\mathcal{A}^{*}(y,x).$ (32h)

To sum up, we found
$\mathcal{A}^{*}(v,u)\in{\operatorname{ran}}\,\mathcal{A}^{*}=(\ker\mathcal{A})^{\perp}$
such that
$\mathcal{A}^{*}\mathcal{A}\mathcal{A}^{*}(v,u)=\mathcal{A}^{*}(y,x)$. By [3,
Proposition 3.30(i)], 30, and 24, we deduce that

$\displaystyle\mathcal{A}^{\dagger}(y,x)$ $\displaystyle=\mathcal{A}^{*}(v,u)$
(33a)
$\displaystyle=\mathcal{A}^{*}\Big{(}\frac{1}{\|e\|^{2}}\big{(}y-\left\langle{f},{{v}}\right\rangle
f\big{)},\frac{1}{\|f\|^{2}}\big{(}x-\left\langle{e},{u}\right\rangle
e\big{)}\Big{)}$ (33b) $\displaystyle=\frac{1}{\|e\|^{2}}\big{(}y\otimes
e-\left\langle{f},{{v}}\right\rangle f\otimes
e\big{)}+\frac{1}{\|f\|^{2}}\big{(}f\otimes x-\left\langle{e},{u}\right\rangle
f\otimes e\big{)}$ (33c)

which now results in 26 by using 28 and 29. $\hfill\quad\blacksquare$

######  Theorem 2.5.

(Moore-Penrose inverse of $\mathcal{A}$ part 2) Let $(y,x)\in Y\times X$. Then
the following hold:

  1. (i)

If $e=0$ and $f\neq 0$, then
$\displaystyle\mathcal{A}^{\dagger}(y,x)=\frac{1}{\|f\|^{2}}f\otimes x$.

  2. (ii)

If $e\neq 0$ and $f=0$, then
$\displaystyle\mathcal{A}^{\dagger}(y,x)=\frac{1}{\|e\|^{2}}y\otimes e$.

  3. (iii)

If $e=0$ and $f=0$, then $\mathcal{A}^{\dagger}(y,x)=0\in\mathcal{H}$.

Proof. Let $T\in\mathcal{H}$.

(i): In this case, $\mathcal{A}(T)=(0,T^{*}f)$ and
$\mathcal{A}^{*}(y,x)=f\otimes x$. Let us verify the Penrose conditions [6,
page 48]. First, using 7,

$\displaystyle\mathcal{A}\mathcal{A}^{\dagger}(y,x)$
$\displaystyle=\mathcal{A}\big{(}\|f\|^{-2}f\otimes
x\big{)}=\|f\|^{-2}\big{(}(f\otimes x)e,(f\otimes x)^{*}f\big{)}$ (34a)
$\displaystyle=\|f\|^{-2}\big{(}0,\left\langle{f},{f}\right\rangle
x\big{)}=(0,x)$ (34b)

and

$\displaystyle\big{\langle}{\mathcal{A}\mathcal{A}^{\dagger}(y,x)},{(v,u)}\big{\rangle}=\big{\langle}{(0,x)},{(v,u)}\big{\rangle}=\left\langle{x},{u}\right\rangle=\big{\langle}{\mathcal{A}\mathcal{A}^{\dagger}(v,u)},{(y,x)}\big{\rangle}$
(35)

which shows that $\mathcal{A}\mathcal{A}^{\dagger}$ is indeed self-adjoint.

Secondly,

$\mathcal{A}^{\dagger}\mathcal{A}(T)=\mathcal{A}^{\dagger}(Te,T^{*}f)=\mathcal{A}^{\dagger}(0,T^{*}f)=\|f\|^{-2}f\otimes(T^{*}f),$
(36)

and if $S\in\mathcal{H}$ and $B$ is any orthonormal basis of $X$, then

$\displaystyle\big{\langle}{\mathcal{A}^{\dagger}\mathcal{A}(T)},{S}\big{\rangle}$
$\displaystyle=\|f\|^{-2}\big{\langle}{f\otimes(T^{*}f)},{S}\big{\rangle}$
(37a) $\displaystyle=\|f\|^{-2}\sum_{b\in
B}\big{\langle}{(f\otimes(T^{*}f))b},{Sb}\big{\rangle}$ (37b)
$\displaystyle=\|f\|^{-2}\sum_{b\in
B}\big{\langle}{\left\langle{T^{*}f},{b}\right\rangle f},{Sb}\big{\rangle}$
(37c) $\displaystyle=\|f\|^{-2}\sum_{b\in
B}\big{\langle}{\left\langle{f},{Tb}\right\rangle f},{Sb}\big{\rangle}$ (37d)
$\displaystyle=\|f\|^{-2}\sum_{b\in
B}\left\langle{f},{Tb}\right\rangle\left\langle{f},{Sb}\right\rangle$ (37e)
$\displaystyle=\big{\langle}{\mathcal{A}^{\dagger}\mathcal{A}(S)},{T}\big{\rangle}$
(37f)

which yields the symmetry of $\mathcal{A}^{\dagger}\mathcal{A}$.

Thirdly, using 36 and the assumption that $e=0$, we have

$\displaystyle\mathcal{A}\mathcal{A}^{\dagger}\mathcal{A}(T)$
$\displaystyle=\mathcal{A}\big{(}\|f\|^{-2}f\otimes(T^{*}f)\big{)}=\|f\|^{-2}\big{(}0,(f\otimes(T^{*}f))^{*}f\big{)}$
(38a) $\displaystyle=\|f\|^{-2}\big{(}0,\left\langle{f},{f}\right\rangle
T^{*}f\big{)}=(0,T^{*}f)$ (38b) $\displaystyle=\mathcal{A}(T).$ (38c)

And finally, using 34, we have

$\mathcal{A}^{\dagger}\mathcal{A}\mathcal{A}^{\dagger}(y,x)=\mathcal{A}^{\dagger}(0,x)=\|f\|^{-2}f\otimes
x=\mathcal{A}^{\dagger}(y,x).$ (39)

(ii): This can be proved similar to (i).

(iii): In this case, $\mathcal{A}$ is the zero operator and hence the Desoer-
Whalen conditions (see [6, page 51]) make it obvious that
$\mathcal{A}^{\dagger}$ is the zero operator as well.
$\hfill\quad\blacksquare$

Let us define the auxiliary function

$\delta(\xi):=\begin{cases}\xi,&\text{if $\xi\neq 0$;}\\\ 1,&\text{if
$\xi=0$}\end{cases}$ (40)

which allows us to combine the previous two results into one:

######  Corollary 2.6.

Let $(y,x)\in Y\times X$. Then

$\displaystyle\mathcal{A}^{\dagger}(y,x)$
$\displaystyle=\frac{1}{\delta(\|e\|^{2})}\Big{(}y\otimes
e-\frac{\left\langle{f},{{y}}\right\rangle}{\delta(\|e\|^{2}+\|f\|^{2})}f\otimes
e\Big{)}$ (41) $\displaystyle\qquad+\frac{1}{\delta(\|f\|^{2})}\Big{(}f\otimes
x-\frac{\left\langle{e},{x}\right\rangle}{\delta(\|e\|^{2}+\|f\|^{2})}f\otimes
e\Big{)}.$ (42)

We now turn to formulas for $P_{{\operatorname{ran}}\,\mathcal{A}}$ and
$P_{{\operatorname{ran}}\,\mathcal{A^{*}}}$.

######  Corollary 2.7.

(projections onto ${\operatorname{ran}}\,\mathcal{A}$ and
${\operatorname{ran}}\,\mathcal{A}^{*}$) Let $(y,x)\in{Y}\times{X}$ and let
$T\in\mathcal{H}$. If $e\neq 0$ and $f\neq 0$, then

$P_{{\operatorname{ran}}\,\mathcal{A}}(y,x)=\mathcal{A}\mathcal{A}^{\dagger}(y,x)=\Big{(}y-\frac{\left\langle{f},{y}\right\rangle-\left\langle{e},{x}\right\rangle}{\|e\|^{2}+\|f\|^{2}}f,x-\frac{\left\langle{e},{x}\right\rangle-\left\langle{f},{y}\right\rangle}{\|e\|^{2}+\|f\|^{2}}e\Big{)}$
(43)

and

$P_{{\operatorname{ran}}\,\mathcal{A}^{*}}(T)=\mathcal{A}^{\dagger}\mathcal{A}(T)=\frac{1}{\|e\|^{2}}(Te)\otimes
e+\frac{1}{\|f\|^{2}}f\otimes(T^{*}f)-\frac{\left\langle{f},{{Te}}\right\rangle}{\|e\|^{2}\|f\|^{2}}f\otimes
e.$ (44)

Furthermore,

$P_{{\operatorname{ran}}\,\mathcal{A}}(y,x)=\mathcal{A}\mathcal{A}^{\dagger}(y,x)=\begin{cases}(0,x),&\text{if
$e=0$ and $f\neq 0$;}\\\\[2.84526pt] (y,0),&\text{if $e\neq 0$ and
$f=0$;}\\\\[2.84526pt] (0,0),&\text{if $e=0$ and $f=0$;}\end{cases}$ (45)

and

$P_{{\operatorname{ran}}\,\mathcal{A}^{*}}(T)=\mathcal{A}^{\dagger}\mathcal{A}(T)=\begin{cases}\displaystyle\frac{1}{\|f\|^{2}}f\otimes(T^{*}f),&\text{if
$e=0$ and $f\neq 0$;}\\\\[11.38109pt]
\displaystyle\frac{1}{\|e\|^{2}}(Te)\otimes e,&\text{if $e\neq 0$ and
$f=0$;}\\\\[11.38109pt] 0,&\text{if $e=0$ and $f=0$.}\end{cases}$ (46)

Proof. Using [3, Proposition 3.30(ii)] and 26, we obtain for $e\neq 0$ and
$f\neq 0$

$\displaystyle P_{{\operatorname{ran}}\,\mathcal{A}}(y,x)$ (47a)
$\displaystyle=\mathcal{A}\mathcal{A}^{\dagger}(y,x)$ (47b)
$\displaystyle=\mathcal{A}\bigg{(}\frac{1}{\|e\|^{2}}\Big{(}y\otimes
e-\frac{\left\langle{f},{{y}}\right\rangle}{\|e\|^{2}+\|f\|^{2}}f\otimes
e\Big{)}+\frac{1}{\|f\|^{2}}\Big{(}f\otimes
x-\frac{\left\langle{e},{x}\right\rangle}{\|e\|^{2}+\|f\|^{2}}f\otimes
e\Big{)}\bigg{)}$ (47c)
$\displaystyle=\bigg{(}\frac{1}{\|e\|^{2}}\Big{(}y\otimes
e-\frac{\left\langle{f},{{y}}\right\rangle}{\|e\|^{2}+\|f\|^{2}}f\otimes
e\Big{)}e+\frac{1}{\|f\|^{2}}\Big{(}f\otimes
x-\frac{\left\langle{e},{x}\right\rangle}{\|e\|^{2}+\|f\|^{2}}f\otimes
e\Big{)}e,$ (47d) $\displaystyle\quad\frac{1}{\|e\|^{2}}\Big{(}y\otimes
e-\frac{\left\langle{f},{{y}}\right\rangle}{\|e\|^{2}+\|f\|^{2}}f\otimes
e\Big{)}^{*}f+\frac{1}{\|f\|^{2}}\Big{(}f\otimes
x-\frac{\left\langle{e},{x}\right\rangle}{\|e\|^{2}+\|f\|^{2}}f\otimes
e\Big{)}^{*}f\bigg{)}$ (47e)
$\displaystyle=\bigg{(}\frac{1}{\|e\|^{2}}\Big{(}\left\langle{e},{e}\right\rangle
y-\frac{\left\langle{f},{{y}}\right\rangle}{\|e\|^{2}+\|f\|^{2}}\left\langle{e},{e}\right\rangle
f\Big{)}+\frac{1}{\|f\|^{2}}\Big{(}\left\langle{x},{e}\right\rangle
f-\frac{\left\langle{e},{x}\right\rangle}{\|e\|^{2}+\|f\|^{2}}\left\langle{e},{e}\right\rangle
f\Big{)},$ (47f)
$\displaystyle\quad\frac{1}{\|e\|^{2}}\Big{(}\left\langle{y},{f}\right\rangle
e-\frac{\left\langle{f},{{y}}\right\rangle}{\|e\|^{2}+\|f\|^{2}}\left\langle{f},{f}\right\rangle
e\Big{)}+\frac{1}{\|f\|^{2}}\Big{(}\left\langle{f},{f}\right\rangle
x-\frac{\left\langle{e},{x}\right\rangle}{\|e\|^{2}+\|f\|^{2}}\left\langle{f},{f}\right\rangle
e\Big{)}\bigg{)}$ (47g)
$\displaystyle=\bigg{(}y-\frac{\left\langle{f},{{y}}\right\rangle}{\|e\|^{2}+\|f\|^{2}}f+\frac{\left\langle{e},{x}\right\rangle}{\|f\|^{2}}f-\frac{\left\langle{e},{x}\right\rangle\|e\|^{2}}{\|f\|^{2}\big{(}\|e\|^{2}+\|f\|^{2}\big{)}}f,$
(47h) $\displaystyle\quad
x-\frac{\left\langle{e},{{x}}\right\rangle}{\|e\|^{2}+\|f\|^{2}}e+\frac{\left\langle{f},{y}\right\rangle}{\|e\|^{2}}e-\frac{\left\langle{f},{y}\right\rangle\|f\|^{2}}{\|e\|^{2}\big{(}\|e\|^{2}+\|f\|^{2}\big{)}}e\bigg{)}$
(47i)
$\displaystyle=\bigg{(}y+\frac{-\left\langle{f},{y}\right\rangle\|f\|^{2}+\left\langle{e},{x}\right\rangle\big{(}\|e\|^{2}+\|f\|^{2}\big{)}-\left\langle{e},{x}\right\rangle\|e\|^{2}}{\|f\|^{2}\big{(}\|e\|^{2}+\|f\|^{2}\big{)}}f,$
(47j) $\displaystyle\quad
x+\frac{-\left\langle{e},{x}\right\rangle\|e\|^{2}+\left\langle{f},{y}\right\rangle\big{(}\|e\|^{2}+\|f\|^{2}\big{)}-\left\langle{f},{y}\right\rangle\|f\|^{2}}{\|e\|^{2}\big{(}\|e\|^{2}+\|f\|^{2}\big{)}}e\bigg{)}$
(47k)
$\displaystyle=\bigg{(}y-\frac{\left\langle{f},{y}\right\rangle-\left\langle{e},{x}\right\rangle}{\|e\|^{2}+\|f\|^{2}}f,x-\frac{\left\langle{e},{x}\right\rangle-\left\langle{f},{y}\right\rangle}{\|e\|^{2}+\|f\|^{2}}e\bigg{)}$
(47l)

which verifies 43.

Next, using [3, Proposition 3.30(v)&(vi)] and 26, we have

$\displaystyle
P_{{\operatorname{ran}}\,\mathcal{A}^{*}}(T)=P_{{\operatorname{ran}}\,\mathcal{A}^{\dagger}}(T)=\mathcal{A}^{\dagger}\mathcal{A}(T)=A^{\dagger}(Te,T^{*}f)$
(48a) $\displaystyle=\frac{1}{\|e\|^{2}}\Big{(}(Te)\otimes
e-\frac{\left\langle{f},{{Te}}\right\rangle}{\|e\|^{2}+\|f\|^{2}}f\otimes
e\Big{)}+\frac{1}{\|f\|^{2}}\Big{(}f\otimes(T^{*}f)-\frac{\left\langle{e},{T^{*}f}\right\rangle}{\|e\|^{2}+\|f\|^{2}}f\otimes
e\Big{)}$ (48b) $\displaystyle=\frac{1}{\|e\|^{2}}(Te)\otimes
e+\frac{1}{\|f\|^{2}}f\otimes(T^{*}f)-\frac{\left\langle{f},{{Te}}\right\rangle}{\|e\|^{2}+\|f\|^{2}}\Big{(}\frac{1}{\|e\|^{2}}+\frac{1}{\|f\|^{2}}\Big{)}f\otimes
e$ (48c) $\displaystyle=\frac{1}{\|e\|^{2}}(Te)\otimes
e+\frac{1}{\|f\|^{2}}f\otimes(T^{*}f)-\frac{\left\langle{f},{{Te}}\right\rangle}{\|e\|^{2}\|f\|^{2}}f\otimes
e$ (48d)

which establishes 44.

If $e=0$ and $f\neq 0$, then

$\mathcal{A}\mathcal{A}^{\dagger}(y,x)=\mathcal{A}(\|f\|^{-2}f\otimes
x)=\|f\|^{-2}(0,(f\otimes
x)^{*}f)=\|f\|^{-2}(0,\left\langle{f},{f}\right\rangle x)=(0,x)$ (49)

and

$\mathcal{A}^{\dagger}\mathcal{A}(T)=\mathcal{A}^{\dagger}(0,T^{*}f)=\frac{1}{\|f\|^{2}}f\otimes(T^{*}f).$
(50)

The case when $e\neq 0$ and $f=0$ is treated similarly.

Finally, if $e=0$ and $f=0$, then $\mathcal{A}^{\dagger}=0$ and the result
follows. $\hfill\quad\blacksquare$

######  Theorem 2.8.

(main projection theorem) Let $(s,r)\in Y\times X$ and set
$(\bar{s},\bar{r})=P_{{\operatorname{ran}}\,\mathcal{A}}(s,r)$. Then

$C:=\mathcal{A}^{-1}(\bar{s},\bar{r})=\big{\\{}{T\in\mathcal{H}}~{}\big{|}~{}{Te=\bar{s}\text{\;and\;}T^{*}f=\bar{r}}\big{\\}}\neq\varnothing.$
(51)

Let $T\in\mathcal{H}$. If $e\neq 0$ and $f\neq 0$, then

$\displaystyle P_{C}(T)$
$\displaystyle=T-\frac{1}{\|e\|^{2}}\Big{(}(Te-s)\otimes
e-\frac{\left\langle{f},{Te-s}\right\rangle}{\|e\|^{2}+\|f\|^{2}}f\otimes
e\Big{)}$ (52a)
$\displaystyle\qquad-\frac{1}{\|f\|^{2}}\Big{(}f\otimes(T^{*}f-r)-\frac{\left\langle{e},{T^{*}f-r}\right\rangle}{\|e\|^{2}+\|f\|^{2}}f\otimes
e\Big{)}.$ (52b)

Moreover,

$P_{C}(T)=\begin{cases}T-\displaystyle\frac{1}{\|f\|^{2}}f\otimes(T^{*}f-r),&\text{if
$e=0$ and $f\neq 0$;}\\\\[8.53581pt]
T-\displaystyle\frac{1}{\|e\|^{2}}(Te-s)\otimes e,&\text{if $e\neq 0$ and
$f=0$;}\\\\[8.53581pt] T,&\text{if $e=0$ and $f=0$.}\end{cases}$ (53)

Proof. Clearly, $C\neq\varnothing$. Now Proposition 1.1 and 11 yield

$\displaystyle P_{C}(T)$
$\displaystyle=T-\mathcal{A}^{\dagger}(\mathcal{A}T-(s,r))=T-\mathcal{A}^{\dagger}(Te-s,T^{*}f-r).$
(54)

Now we consider all possible cases. If $e\neq 0$ and $f\neq 0$, then, using
26,

$\displaystyle P_{C}(T)$
$\displaystyle=T-\frac{1}{\|e\|^{2}}\Big{(}(Te-s)\otimes
e-\frac{\left\langle{f},{{Te-s}}\right\rangle}{\|e\|^{2}+\|f\|^{2}}f\otimes
e\Big{)}$
$\displaystyle\qquad-\frac{1}{\|f\|^{2}}\Big{(}f\otimes(T^{*}f-r)-\frac{\left\langle{e},{T^{*}f-r}\right\rangle}{\|e\|^{2}+\|f\|^{2}}f\otimes
e\Big{)}$

as claimed.

Next, if $e=0$ and $f\neq 0$, then using Theorem 2.5(i) yields

$\displaystyle P_{C}(T)$
$\displaystyle=T-\frac{1}{\|f\|^{2}}f\otimes(T^{*}f-r).$

Similarly, if $e\neq 0$ and $f=0$, then using Theorem 2.5(ii) yields

$\displaystyle P_{C}(T)$ $\displaystyle=T-\frac{1}{\|e\|^{2}}(Te-s)\otimes e.$

And finally, if $e=0$ and $f=0$, then $\mathcal{A}^{\dagger}=0$ and hence
$P_{C}(T)=T$. $\hfill\quad\blacksquare$

######  Remark 2.9.

Consider Theorem 2.8 and its notation. If
$(s,r)\in{\operatorname{ran}}\,\mathcal{A}$, then $(\bar{s},\bar{r})=(s,r)$
and hence $C=\mathcal{A}^{-1}(s,r)$ which covers also the consistent case.
Note that the auxiliary function defined in 40 allows us to combine all four
cases into

$\displaystyle P_{C}(T)$
$\displaystyle=T-\frac{1}{\delta(\|e\|^{2})}\Big{(}(Te-s)\otimes
e-\frac{\left\langle{f},{Te-s}\right\rangle}{\delta(\|e\|^{2}+\|f\|^{2})}f\otimes
e\Big{)}$ (55a)
$\displaystyle\qquad-\frac{1}{\delta(\|f\|^{2})}\Big{(}f\otimes(T^{*}f-r)-\frac{\left\langle{e},{T^{*}f-r}\right\rangle}{\delta(\|e\|^{2}+\|f\|^{2})}f\otimes
e\Big{)}.$ (55b)

The last two results in this section are inspired by [5, Theorem 2.1] and [8,
Theorem on page 566], respectively. See also Corollary 3.6 and Example 3.8
below.

######  Corollary 2.10.

Suppose that $Y=X$, let $e\in X\smallsetminus\\{0\\}$, let $f\in
X\smallsetminus\\{0\\}$, set

$E:=\frac{1}{\|e\|^{2}}e\otimes e=P_{\,\mathbb{R}e}\quad\text{and}\quad
F:=\frac{1}{\|f\|^{2}}f\otimes f=P_{\,\mathbb{R}f},$ (56)

and let $\gamma\in\mathbb{R}$. Then

$C:=\big{\\{}{T\in\mathcal{H}}~{}\big{|}~{}{Te=\gamma
e\;\text{and}\;T^{*}f=\gamma f}\big{\\}}\neq\varnothing$ (57)

and

$(\forall T\in\mathcal{H})\quad
P_{C}(T)=\gamma\operatorname{Id}+(\operatorname{Id}-F)(T-\gamma\operatorname{Id})(\operatorname{Id}-E).$
(58)

Proof. The projection identities in 56 follow from 9. Note that
$\gamma\operatorname{Id}\in C$ and hence $C\neq\varnothing$. We may and do
assume without loss of generality that $\|e\|=1=\|f\|$.

Now let $T\in\mathcal{H}$. Applying 52 with $r:=\gamma f$ and $s:=\gamma e$,
we deduce that

$\displaystyle P_{C}(T)$ $\displaystyle=T-\Big{(}(Te-\gamma e)\otimes
e-\frac{\left\langle{f},{Te-\gamma e}\right\rangle}{2}f\otimes e\Big{)}$ (59a)
$\displaystyle\qquad-\Big{(}f\otimes(T^{*}f-\gamma
f)-\frac{\left\langle{e},{T^{*}f-\gamma f}\right\rangle}{2}f\otimes e\Big{)}$
(59b) $\displaystyle=T-(Te)\otimes e+\gamma e\otimes
e+\frac{\left\langle{f},{Te}\right\rangle-\gamma\left\langle{f},{e}\right\rangle}{2}f\otimes
e$ (59c) $\displaystyle\qquad-f\otimes(T^{*}f)+\gamma f\otimes
f+\frac{\left\langle{Te},{f}\right\rangle-\gamma\left\langle{e},{f}\right\rangle}{2}f\otimes
e$ (59d) $\displaystyle=T-(Te)\otimes
e-f\otimes(T^{*}f)+\gamma(E+F)+\big{(}\left\langle{f},{Te}\right\rangle-\gamma\left\langle{e},{f}\right\rangle\big{)}f\otimes
e$ (59e) $\displaystyle=T-TE-
FT+\gamma(E+F)+\big{(}\left\langle{f},{Te}\right\rangle-\gamma\left\langle{e},{f}\right\rangle\big{)}f\otimes
e$ (59f) $\displaystyle=T-TE-FT+\gamma(E+F)+FTE-\gamma FE$ (59g)
$\displaystyle=\gamma\operatorname{Id}+T-TE-
FT+FTE-\gamma\operatorname{Id}+\gamma E+\gamma F-\gamma FE$ (59h)
$\displaystyle=\gamma\operatorname{Id}+(\operatorname{Id}-F)T(\operatorname{Id}-E)-\gamma(\operatorname{Id}-F)(\operatorname{Id}-E)$
(59i)
$\displaystyle=\gamma\operatorname{Id}+(\operatorname{Id}-F)(T-\gamma\operatorname{Id})(\operatorname{Id}-E)$
(59j) as claimed.

$\hfill\quad\blacksquare$

We conclude this section with a beautiful projection formula that arises when
the last result is specialized even further.

######  Corollary 2.11.

Suppose that $Y=X$, let $e\in X\smallsetminus\\{0\\}$, and set

$E:=\frac{1}{\|e\|^{2}}e\otimes e=P_{\,\mathbb{R}e}.$ (60)

Then

$C:=\big{\\{}{T\in\mathcal{H}}~{}\big{|}~{}{Te=e=T^{*}e}\big{\\}}\neq\varnothing$
(61)

and

$(\forall T\in\mathcal{H})\quad
P_{C}(T)=E+(\operatorname{Id}-E)T(\operatorname{Id}-E).$ (62)

Proof. Let $T\in\mathcal{H}$. Applying Corollary 2.10 with $f=e$ and
$\gamma=1$, we obtain

$\displaystyle P_{C}(T)$
$\displaystyle=\operatorname{Id}+(\operatorname{Id}-E)(T-\operatorname{Id})(\operatorname{Id}-E)$
(63a)
$\displaystyle=\operatorname{Id}+(\operatorname{Id}-E)T(\operatorname{Id}-E)-(\operatorname{Id}-E)^{2}$
(63b)
$\displaystyle=\operatorname{Id}+(\operatorname{Id}-E)T(\operatorname{Id}-E)-(\operatorname{Id}-E)$
(63c) $\displaystyle=(\operatorname{Id}-E)T(\operatorname{Id}-E)+E$ (63d)

because $\operatorname{Id}-E=P_{\\{e\\}^{\perp}}$ is idempotent.
$\hfill\quad\blacksquare$

##  3 Rectangular matrices

In this section, we specialize the results of Section 2 to

$X=\mathbb{R}^{n}$ and $Y=\mathbb{R}^{m}$, (64)

which gives rise to

$\mathcal{H}=\mathbb{R}^{m\times n},$ (65)

the space of real $m\times n$ matrices. Given $u$ and $x$ in $\mathbb{R}^{n}$,
and $v$ and $y$ in $\mathbb{R}^{m}$, we have $v\otimes u=vu^{\intercal}$,
$(v\otimes u)x=vu^{\intercal}x=(u^{\intercal}x)v$, and $(v\otimes
u)^{*}y=(v^{\intercal}y)u$. Corresponding to 11, we have

$\mathcal{A}\colon\mathbb{R}^{m\times n}\to\mathbb{R}^{m+n}\colon
T\mapsto\begin{bmatrix}Te\\\ T^{\intercal}f\end{bmatrix}.$ (66)

The counterpart of 24 reads

$\mathcal{A}^{*}\colon\mathbb{R}^{m+n}\to\mathbb{R}^{m\times
n}\colon\begin{bmatrix}y\\\ x\end{bmatrix}\mapsto
ye^{\intercal}+fx^{\intercal}.$ (67)

Translated to the matrix setting, Theorem 2.4 and Theorem 2.5 turn into:

######  Theorem 3.1.

Let $x\in\mathbb{R}^{n}$ and $y\in\mathbb{R}^{m}$. If $e\neq 0$ and $f\neq 0$,
then

$\mathcal{A}^{\dagger}\begin{bmatrix}y\\\
x\end{bmatrix}=\frac{1}{\|e\|^{2}}\Big{(}ye^{\intercal}-\frac{{f}^{\intercal}{{y}}}{\|e\|^{2}+\|f\|^{2}}fe^{\intercal}\Big{)}+\frac{1}{\|f\|^{2}}\Big{(}fx^{\intercal}-\frac{{e}^{\intercal}{x}}{\|e\|^{2}+\|f\|^{2}}fe^{\intercal}\Big{)}.$
(68)

Furthermore,

$\mathcal{A}^{\dagger}\begin{bmatrix}y\\\
x\end{bmatrix}=\begin{cases}\displaystyle\frac{1}{\|f\|^{2}}fx^{\intercal},&\text{if
$e=0$ and $f\neq 0$;}\\\\[11.38109pt]
\displaystyle\frac{1}{\|e\|^{2}}ye^{\intercal},&\text{if $e\neq 0$ and
$f=0$;}\\\\[11.38109pt] 0,&\text{if $e=0$ and $f=0$.}\end{cases}$ (69)

In turn, Corollary 2.7 now states the following:

######  Corollary 3.2.

Let $x\in\mathbb{R}^{n}$, let $y\in\mathbb{R}^{m}$, and let
$T\in\mathbb{R}^{m\times n}$. If $e\neq 0$ and $f\neq 0$, then

$P_{{\operatorname{ran}}\,\mathcal{A}}\begin{bmatrix}y\\\
x\end{bmatrix}=\begin{bmatrix}y\\\
x\end{bmatrix}-\frac{f^{\intercal}y-e^{\intercal}x}{\|e\|^{2}+\|f\|^{2}}\begin{bmatrix}f\\\
-e\end{bmatrix}$ (70)

and

$P_{{\operatorname{ran}}\,\mathcal{A}^{*}}(T)=\frac{1}{\|e\|^{2}}Tee^{\intercal}+\frac{1}{\|f\|^{2}}ff^{\intercal}T-\frac{f^{\intercal}Te}{\|e\|^{2}\|f\|^{2}}fe^{\intercal}.$
(71)

Furthermore,

$P_{{\operatorname{ran}}\,\mathcal{A}}\begin{bmatrix}y\\\
x\end{bmatrix}=\begin{cases}\begin{bmatrix}0\\\\[-5.69054pt]
x\end{bmatrix},&\text{if $e=0$ and $f\neq 0$;}\\\\[11.38109pt]
\begin{bmatrix}y\\\\[-2.84526pt] 0\end{bmatrix},&\text{if $e\neq 0$ and
$f=0$;}\\\\[11.38109pt] \begin{bmatrix}0\\\\[-5.69054pt]
0\end{bmatrix},&\text{if $e=0$ and $f=0$;}\end{cases}$ (72)

and

$P_{{\operatorname{ran}}\,\mathcal{A}^{*}}(T)=\begin{cases}\displaystyle\frac{1}{\|f\|^{2}}ff^{\intercal}T,&\text{if
$e=0$ and $f\neq 0$;}\\\\[11.38109pt]
\displaystyle\frac{1}{\|e\|^{2}}Tee^{\intercal},&\text{if $e\neq 0$ and
$f=0$;}\\\\[11.38109pt] 0,&\text{if $e=0$ and $f=0$.}\end{cases}$ (73)

Next, Theorem 2.8 turns into the following result:

######  Theorem 3.3.

Let $r\in\mathbb{R}^{n}$, let $s\in\mathbb{R}^{m}$, and set
$[\bar{s},\bar{r}]^{\intercal}=P_{{\operatorname{ran}}\,\mathcal{A}}[s,r]^{\intercal}$.
Then

$C:=\big{\\{}{T\in\mathbb{R}^{m\times
n}}~{}\big{|}~{}{Te=\bar{s}\;\text{and}\;T^{\intercal}f=\bar{r}}\big{\\}}\neq\varnothing.$
(74)

Now let $T\in\mathbb{R}^{m\times n}$. If $e\neq 0$ and $f\neq 0$, then

$\displaystyle P_{C}(T)$
$\displaystyle=T-\frac{1}{\|e\|^{2}}\Big{(}(Te-s)e^{\intercal}-\frac{{f^{\intercal}}(Te-s)}{\|e\|^{2}+\|f\|^{2}}fe^{\intercal}\Big{)}$
(75a)
$\displaystyle\qquad-\frac{1}{\|f\|^{2}}\Big{(}f(f^{\intercal}T-r^{\intercal})-\frac{{e^{\intercal}}{(T^{\intercal}f-r)}}{\|e\|^{2}+\|f\|^{2}}fe^{\intercal}\Big{)}.$
(75b)

Moreover,

$P_{C}(T)=\begin{cases}T-\displaystyle\frac{1}{\|f\|^{2}}f(f^{\intercal}T-r^{\intercal}),&\text{if
$e=0$ and $f\neq 0$;}\\\\[8.53581pt]
T-\displaystyle\frac{1}{\|e\|^{2}}(Te-s)e^{\intercal},&\text{if $e\neq 0$ and
$f=0$;}\\\\[8.53581pt] T,&\text{if $e=0$ and $f=0$.}\end{cases}$ (76)

Let us specialize Theorem 3.3 further to the following interesting case:

######  Corollary 3.4.

(projection onto matrices with prescribed row/column sums)
Suppose that $e=[1,1,\ldots,1]^{\intercal}\in\mathbb{R}^{n}$ and that
$f=[1,1,\ldots,1]^{\intercal}\in\mathbb{R}^{m}$. Let $r\in\mathbb{R}^{n}$, let
$s\in\mathbb{R}^{m}$, and set
$[\bar{s},\bar{r}]^{\intercal}=P_{{\operatorname{ran}}\,\mathcal{A}}[s,r]^{\intercal}$.
Then

$C:=\big{\\{}{T\in\mathbb{R}^{m\times
n}}~{}\big{|}~{}{Te=\bar{s}\;\text{and}\;T^{\intercal}f=\bar{r}}\big{\\}}\neq\varnothing,$
(77)

and for every $T\in\mathbb{R}^{m\times n}$,

$\displaystyle P_{C}(T)$
$\displaystyle=T-\frac{1}{n}\Big{(}(Te-s)e^{\intercal}-\frac{{f^{\intercal}}(Te-s)}{n+m}fe^{\intercal}\Big{)}$
(78a)
$\displaystyle\qquad-\frac{1}{m}\Big{(}f(f^{\intercal}T-r^{\intercal})-\frac{{e^{\intercal}}{(T^{\intercal}f-r)}}{n+m}fe^{\intercal}\Big{)}.$
(78b)

######  Remark 3.5.

(Romero; 1990) Consider Corollary 3.4 and its notation. Assume that
$[s,r]^{\intercal}\in{\operatorname{ran}}\,\mathcal{A}$, which is equivalent
to requiring that
$\left\langle{e},{r}\right\rangle=\left\langle{f},{s}\right\rangle$ (which is
sometimes jokingly called the “Fundamental Theorem of Accounting”). Then one
verifies that the entries of the matrix in 78 are given also expressed by

$\big{(}P_{C}(T)\big{)}_{i,j}=T_{i,j}+\frac{s_{i}-(Te)_{i}}{n}+\frac{r_{j}-(T^{\intercal}f)_{j}}{m}+\frac{{f^{\intercal}}{Te}-{e^{\intercal}}{r}}{mn}$
(79)

for every $i\in\\{1,\ldots,m\\}$ and $j\in\\{1,\ldots,n\\}$. Formula 79 was
proved by Romero (see [11, Corollary 2.1]) who proved this result using
Lagrange multipliers and who has even a $K$-dimensional extension (where 79
corresponds to $K=2$). We also refer the reader to [4] for using 79 to compute
the projection onto the transportation polytope.

Next, Corollary 2.10 turns into the following result:

######  Corollary 3.6.

(Glunt-Hayden-Reams; 1998) [5, Theorem 2.1] Suppose that $e$ and $f$ lie in
$\mathbb{R}^{n}\smallsetminus\\{0\\}$, set

$E:=\frac{1}{\|e\|^{2}}ee^{\intercal}\quad\text{and}\quad
F:=\frac{1}{\|f\|^{2}}ff^{\intercal},$ (80)

and let $\gamma\in\mathbb{R}$. Then

$C:=\big{\\{}{T\in\mathbb{R}^{n\times n}}~{}\big{|}~{}{Te=\gamma
e\;\text{and}\;T^{\intercal}f=\gamma f}\big{\\}}\neq\varnothing$ (81)

and

$(\forall T\in\mathcal{H})\quad
P_{C}(T)=\gamma\operatorname{Id}+(\operatorname{Id}-F)(T-\gamma\operatorname{Id})(\operatorname{Id}-E).$
(82)

We conclude this section with a particularization of Corollary 2.11 which
immediately follows when $X=Y=\mathbb{R}^{n}$ and thus
$\mathcal{H}=\mathbb{R}^{n\times n}$:

######  Corollary 3.7.

Suppose that $e\in\mathbb{R}^{n}\smallsetminus\\{0\\}$, and set

$E:=\frac{1}{\|e\|^{2}}ee^{\intercal}.$ (83)

Then

$C:=\big{\\{}{T\in\mathbb{R}^{n\times
n}}~{}\big{|}~{}{Te=e=T^{\intercal}e}\big{\\}}\neq\varnothing$ (84)

and

$(\forall T\in\mathbb{R}^{n\times n})\quad
P_{C}(T)=E+(\operatorname{Id}-E)T(\operatorname{Id}-E).$ (85)

######  Example 3.8.

(projection formula for generalized bistochastic matrices; 1998)
(See [8, Theorem on page 566] and [5, Corollary 2.1].) Set

$u:=[1,1,\ldots,1]^{\intercal}\in\mathbb{R}^{n},\;\;C:=\big{\\{}{T\in\mathbb{R}^{n\times
n}}~{}\big{|}~{}{Tu=u=T^{\intercal}u}\big{\\}},\;\;\text{and}\;\;J:=(1/n)uu^{\intercal}.$
(86)

Then

$(\forall T\in\mathbb{R}^{n\times n})\quad
P_{C}(T)=J+(\operatorname{Id}-J)T(\operatorname{Id}-J).$ (87)

Proof. Apply Corollary 3.7 with $e=u$ for which $\|e\|^{2}=n$.
$\hfill\quad\blacksquare$

######  Remark 3.9.

For some applications of Example 3.8, we refer the reader to [12] and also to
the recent preprint [2].

######  Remark 3.10.

A reviewer pointed out that projection algorithms can also be employed to
solve linear programming problems provided a strict complementary condition
holds (see Nurminski’s work [9]). This does suggest a possibly interesting
future project: explore whether the projections in this paper are useful in
solving some linear programming problems on rectangular matrices with
prescribed row and column sums.

##  4 Numerical experiments

We consider the problem of finding a rectangular matrix with prescribed row
and column sums as well as some additional constraints on the entries of the
matrix. To be specific and inspired by [1], we seek a real matrix of size
$m\times n=4\times 5$ such that its row and column sums are equal to
$\bar{s}:=\begin{bmatrix}32,43,33,23\end{bmatrix}^{\intercal}$ and
$\bar{r}:=\begin{bmatrix}24,18,37,27,25\end{bmatrix}^{\intercal}$,
respectively. One solution featuring actually nonnegative integers to this
problem is given by

$\begin{array}[]{rrrrr|r}9&4&8&4&7&32\\\ 7&9&15&7&5&43\\\ 3&2&9&10&9&33\\\
5&3&5&6&4&23\\\ \hline\cr 24&18&37&27&25&131\end{array}$

Adopting the notation of Corollary 3.4, we see that the set

$B:=\big{\\{}{T\in\mathbb{R}^{4\times
5}}~{}\big{|}~{}{Te=\bar{s}\;\text{and}\;T^{\intercal}f=\bar{r}}\big{\\}}\neq\varnothing$
(88)

is an affine subspace of $\mathbb{R}^{4\times 5}$ and that an explicit formula
for $P_{B}$ is available through Corollary 3.4. Next, we define the closed
convex “hyper box”

$A:=\bigtimes_{\begin{subarray}{c}i\in\\{1,2,3,4\\}\\\
j\in\\{1,2,3,4,5\\}\end{subarray}}\big{[}0,\min\\{{\bar{s}}_{i},{\bar{r}}_{j}\\}\big{]}.$
(89)

For instance, the $(1,3)$-entry of any nonnegative integer solution must lie
between $0$ and $32=\min\\{32,37\\}$; thus $A_{1,3}=[0,32]$. The projection of
a real number $\xi$ onto the interval $[0,\min({\bar{s}}_{i},{\bar{r}}_{j})]$
is given by $\max\\{0,\min\\{{\bar{s}}_{i},{\bar{r}}_{j},\xi\\}\\}$. Because
$A$ is the Cartesian product of such intervals, the projection operator
$P_{A}$ is nothing but the corresponding product of interval projection
operators.

Our problem is thus to

find a matrix $T$ in $A\cap B$. (90)

We shall tackle 90 with three well known algorithms: Douglas-Rachford (DR),
Method of Alternating Projections (MAP), and Dykstra (Dyk). Here is a quick
review of how these methods operate, for a given starting matrix
$T_{0}\in\mathbb{R}^{4\times 5}$ and a current matrix
$T_{k}\in\mathbb{R}^{4\times 5}$.

DR updates via

$T_{k+1}:=T_{k}-P_{A}(T_{k})+P_{B}(2P_{A}(T_{k})-T_{k}),$ (DR)

MAP updates via

$T_{k+1}:=P_{B}(P_{A}(T_{k})),$ (MAP)

and finally Dyk initializes also $R_{0}=0\in\mathbb{R}^{4\times 5}$ and
updates via

$A_{k+1}:=P_{A}(T_{k}+R_{k}),\;\;R_{k+1}:=T_{k}+R_{k}-A_{k+1},\;\;T_{k+1}:=P_{B}(A_{k+1}).$
(Dyk)

For all three algorithms, it is known that

$P_{A}(T_{k})\to\text{some matrix in $A\cap B$;}$ (91)

in fact, Dyk satisfies even $P_{A}(T_{k})\to P_{A\cap B}(T_{0})$ (see, e.g.,
[3, Corollary 28.3, Corollary 5.26, and Theorem 30.7]). Consequently, for each
of the three algorithms, we will focus on the sequence

$(P_{A}(T_{k}))_{k\in\mathbb{N}}$ (92)

which obviously lies in $A$ and which thus prompts the simple feasibility
criterion given by

$\delta_{k}:=\|P_{A}(T_{k})-P_{B}(P_{A}(T_{k}))\|.$ (93)

###  4.1 The convex case

Figure 1: Convergence of iterates with the nonnegative matrix constraint

Each algorithm is run for 250 iterations and for $100,000$ instances of
$T_{0}$ that are produced with entries generated uniformly in $[-100,100]$.
The plot of the median value for $\delta_{k}$ of the iterates is shown in Fig.
1. The shaded region for each line represents the range of values attained at
that iteration. We assume an algorithm to have achieved feasibility when
$\delta_{k}=0$. While MAP and DR always achieve feasibility, as can be seen
from the range of their values in Fig. 1, DR achieves it the fastest in most
cases. To support this, we order these algorithms in Table 1 according to
their performance. The first column reports what percent of the instances
achieved feasibility in the given order and if any of the algorithms did not
converge. So the row labelled “DR$<$MAP” represents cases where DR achieved
feasibility the fastest, MAP was second and Dyk did not converge. The second
column report what percent of the first feasible matrices obtained were
closest to the starting point $T_{0}$ in the given order. This is done by
measuring $\left\lVert T_{0}-T\right\rVert$, where
$\left\lVert\cdot\right\rVert$ is the operator norm, and $T_{k}$ is the first
feasible matrices obtained using a given algorithm (Dyk, DR or MAP) We
consider the algorithms tied, if the distance between the starting point and
the estimate for both differs by a value less than or equal to $10^{-15}$. As
is evident, a majority of the cases have DR in the lead for feasibility.
However, the distance of these matrices is not as close as the ones given by
MAP and Dyk when feasible. This is consistent with the fact that DR explores
regions further away from the starting point to look for matrices and Dyk is
built to achieve the least distance. It’s also worth noting that at least one
of these algorithms converges in every instance. (Convergence for all three
algorithms is guaranteed in theory.)

Last but not least, because our problem deals with _unscaled_ row and column
sums, we point out that the sought-after projection may also be computed by
using the algorithm proposed by Calvillo and Romero [4] which even converges
in finitely many steps!

Outcome |  | using iterations
---
for feasibility
| using distance
---
from $T_{0}$
DR$=$MAP | 3 | -
DR$=$MAP$<$Dyk | 21 | -
DR$<$MAP | 21,205 | -
MAP$<$DR | 2 | 21,210
DR$<$MAP$=$Dyk | 3 | -
DR$<$MAP$<$Dyk | 78,708 | -
DR$<$Dyk$<$MAP | 11 | -
MAP$<$DR$<$Dyk | 47 | -
Dyk$<$DR$<$MAP | - | 11
Dyk$<$MAP$<$DR | - | 78,779
Total | 100,000 | 100,000
Table 1: Results for nonnegative matrices

###  4.2 The nonconvex case

We exactly repeat the experiment of Section 4.1 with the only difference being
that the (new) set $A$ in this section is the intersection of the (old) set
$A$ from the previous section (see 89) and $\mathbb{Z}^{4\times 5}$. This
enforces nonnegative integer solutions. The projection operator $P_{A}$ is
obtained by simply rounding after application of $P_{A}$ from Section 4.1.

In this nonconvex case, MAP fails to converge in most cases, whereas DR and
Dyk converge to solutions as shown in Fig. 2. This is corroborated by Table 2
where the rows where MAP converges corresponds to only a quarter of the total
cases. Again, DR achieves feasibility the fastest in more than half the cases,
but Dykstra’s algorithm gives the solution closest to $T_{0}$ among these, as
shown in the second column of Table 2. In this nonconvex case convergence of
the any of the algorithms is not guaranteed; in fact, there are several
instances when no solution is found. However, in the $10^{5}$ runs considered,
we did end up discovering several distinct solutions (see Table 3). It turned
out that all solutions found were distinct even across all three algorithms
resulting in 113622 different nonnegative integer solutions in total.

Figure 2: Convergence of iterates with the integer matrix constraint Outcome |  | using iterations
---
for feasibility
| using distance
---
from $T_{0}$
None | 11,694 | 11,694
DR | 19,468 | 19,468
MAP | 25,380 | 25,380
Dyk | 45 | 45
DR$=$Dyk | 164 | -
DR$<$Dyk | 41,822 | -
MAP$<$DR | 73 | 73
MAP$<$Dyk | 69 | -
Dyk$<$DR | 1156 | 43,142
Dyk$<$MAP | - | 69
MAP$<$Dyk$<$DR | 1 | -
MAP$<$DR$<$Dyk | 128 | -
Dyk$<$MAP$<$DR | - | 129
Total | 100,000 | 100,000
Table 2: Results for nonnegative integer matrices  Algorithm | Solutions found | Unique cases
---|---|---
DR | 62,812 | 62,812
MAP | 316 | 314
Dyk | 68,725 | 50,496
Total | 131,853 | 113,622
Table 3: Integer matrix solutions found by the three algorithms

Declaration

Availability of data and materials The datasets generated during and/or
analysed during the current study are available from the corresponding author
on reasonable request.
Competing interests The authors declare that they have no competing interests.
Funding The research of HHB and XW was partially supported by Discovery Grants
from the Natural Sciences and Engineering Research Council of Canada.
Authors’ contributions All authors contributed equally in writing this
article. All authors read and approved the final manuscript.
Authors’ acknowledgments We thank the editor Aviv Gibali, three anonymous
reviewers, and Matt Tam for constructive comments and several pointers to
literature we were previously unaware of.

## References

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††thanks<EMAIL_ADDRESS>ORCID: 0000-0002-8850-7580††thanks:
<EMAIL_ADDRESS>ORCID: 0000-0001-8837-4127

# The Dark Matter Discovery Potential of the
Advanced Particle-Astrophysics Telescope (APT)

Fei Xua,b Dan Hoopera,b,c aUniversity of Chicago, Department of Astronomy and
Astrophysics, Chicago, Illinois 60637, USA  bUniversity of Chicago, Kavli
Institute for Cosmological Physics, Chicago, Illinois 60637, USA  cFermi
National Accelerator Laboratory, Theoretical Astrophysics Group, Batavia,
Illinois 60510, USA

###### Abstract

Gamma-ray observations of Milky Way dwarf galaxies have been used to place
stringent constraints on the dark matter’s annihilation cross section. In this
paper, we evaluate the sensitivity of the proposed Advanced Particle-
astrophysics Telescope (APT) to dark matter in these systems, finding that
such an instrument would be capable of constraining thermal relics with masses
as large as $m_{X}\sim 600\,{\rm GeV}$. Furthermore, in dark matter scenarios
motivated by the observed Galactic Center Gamma-Ray Excess, we predict that
APT would detect several dwarf galaxies with high-significance. Such
observations could be used to test the predicted proportionality between the
gamma-ray fluxes and $J$-factors of individual dwarf galaxies, providing us
with an unambiguous test of the origin of the Galactic Center Excess.

††preprint: FERMILAB-PUB-23-XXX-T††preprint: FERMILAB-PUB-23-462-T

##  I Introduction

For decades, the most widely studied dark matter candidates have been thermal
relics of the early universe [1]. Such particles were motivated by the
realization that, if they had weak-scale masses and couplings, they would
freeze out of equilibrium with an abundance that is approximately equal to
that of the measured dark matter density – a fact frequently referred to as
the “WIMP miracle.” Many WIMP models, however, have since been ruled out by
the null results of direct [2, 3] and indirect [4, 5, 6, 7] searches. In light
of this progress, some have argued that the WIMP paradigm is now disfavored.
Simultaneously, it is not difficult to identify WIMP models which continue to
be consistent with all existing constraints (for a discussion, see Ref. [8]).

The WIMP paradigm has been bolstered in recent years by the signal known as
the Galactic Center Gamma-Ray Excess, as identified within the data collected
by the Fermi Gamma-Ray Space Telescope [9, 10] (for early work, see Refs. [11,
12, 13, 14, 15, 16, 17, 18, 19]). The spectrum and angular distribution of
this excess are each consistent with those predicted from annihilating dark
matter. In particular, the measured characteristics of this signal are well
fit by dark matter particles with a mass of $m_{X}\sim 35-60\,{\rm GeV}$ and
an annihilation cross section of $\langle\sigma v\rangle\sim(1-3)\times
10^{-26}\,{\rm cm}^{3}/{\rm s}$ (for the case of annihilations to $b\bar{b}$).
This agrees remarkably well with the annihilation cross section expected of a
thermal relic, $\langle\sigma v\rangle\approx 2.2\times 10^{-26}\,{\rm
cm}^{3}/{\rm s}$ (see, for example, Ref. [20]).

If the Galactic Center Gamma-Ray Excess is generated by annihilating dark
matter, an analogous signal should be produced from dwarf galaxies (for a
review, see Ref. [21]). Gamma-ray observations of the Milky Way’s dwarf galaxy
population thus have the potential to confirm or rule out dark matter
interpretations of the Galactic Center excess and, more broadly, to test the
WIMP paradigm itself. The most recent such analysis by the Fermi Collaboration
studied six years of data from the directions of 45 dwarf galaxies (and dwarf
galaxy candidates), allowing them to rule out dark matter annihilation cross
sections greater than $\langle\sigma v\rangle\sim 2.2\times 10^{-26}\,{\rm
cm}^{3}/{\rm s}$ for masses up to $m_{X}\sim 50\,{\rm GeV}$ (again, for the
case of annihilations to $b\bar{b}$) [4]. More recently, Di Mauro et al. used
14 years of Fermi data to study a sample of 22 dwarf galaxies, producing
similar constraints [22]. Perhaps more interesting, these analyses also
identified what could be hints of a dark matter annihilation signal. In
particular, Ref. [22] reports the presence of gamma-ray excesses from the
dwarf galaxies Reticulum II, Sculptor, and Tucana II at a level of ${\rm
TS}\approx 11$, 9, and 6, respectively (for the case of $m_{X}=50\,{\rm
GeV}$), where TS is the log-likelihood test statistic. Overall, this stacked
analysis favors the presence of annihilating dark matter over the null
hypothesis at a level of ${\rm TS}\approx 11$, corresponding to a local
significance of 3.0$\sigma$ [22] (see also, Refs. [23, 24, 25]).

Whereas gamma-ray studies of the Galactic Center are currently limited by
systematic uncertainties associated with bright and poorly understood
backgrounds, searches for gamma rays from dwarf galaxies are statistically
limited. Future observations of the Milky Way’s dwarf galaxy population with a
large-acceptance, space-based gamma-ray telescope could thus significantly
increase our sensitivity to dark matter and clarify the origin of the Galactic
Center Gamma-Ray Excess.

Dwarf Galaxy | Distance (kpc) | $\log_{10}J(0.5\degree)$ |  $l$ ($\degree$) |  $b$ ($\degree$) | Included in Fig. 1
---|---|---|---|---|---
Canes Venatici I |  210.0 $\pm$ 6.0 | $17.42_{-0.15}^{+0.17}$ | 74.30 | 79.83 | No
Carina |  105.6 $\pm$ 5.4 | $17.83_{-0.09}^{+0.10}$ | 260.11 | -22.22 | Yes
Draco |  76.0 $\pm$ 6.0 | $18.83_{-0.12}^{+0.12}$ | 86.37 | 34.71 | Yes
Fornax |  147.0 $\pm$ 9.0 | $18.09_{-0.10}^{+0.10}$ | 237.24 | -65.67 | Yes
Leo I |  258.2 $\pm$ 9.5 | $17.64_{-0.12}^{+0.14}$ | 225.98 | 49.11 | No
Leo II |  233.0 $\pm$ 15.0 | $17.76_{-0.18}^{+0.22}$ | 220.16 | 67.23 | Yes
Sculptor |  83.9 $\pm$ 1.5 | $18.58_{-0.05}^{+0.05}$ | 287.70 | -83.15 | Yes
Sextans |  92.5 $\pm$ 2.2 | $17.73_{-0.12}^{+0.13}$ | 243.50 | 42.27 | Yes
Ursa Minor |  76.0 $\pm$ 4.0 | $18.75_{-0.12}^{+0.12}$ | 104.98 | 44.81 | Yes
Aquarius II |  107.9 $\pm$ 3.3 | $18.27_{-0.58}^{+0.66}$ | 55.11 | -53.01 | No
Boötes I |  66.0 $\pm$ 3.0 | $18.17_{-0.29}^{+0.31}$ | 358.10 | 69.64 | Yes
Canes Venatici II |  160.0 $\pm$ 7.0 | $17.82_{-0.47}^{+0.47}$ | 113.57 | 82.70 | Yes
Carina II |  37.4 $\pm$ 0.4 | $18.25_{-0.54}^{+0.55}$ | 269.98 | -17.14 | No
Coma Berenices |  42.0 $\pm$ 1.5 | $19.00_{-0.35}^{+0.36}$ | 241.86 | 83.61 | Yes
Draco II |  20.0 $\pm$ 3.0 | $18.93_{-1.70}^{+1.39}$ | 98.32 | 42.88 | No
Grus I |  120.2 $\pm$ 11.1 | $16.88_{-1.66}^{+1.51}$ | 338.65 | -58.24 | No
Hercules |  132.0 $\pm$ 6.0 | $17.37_{-0.53}^{+0.53}$ | 28.73 | 36.86 | Yes
Horologium I B |  87.0 $\pm$ 8.0 | $18.79_{-0.86}^{+0.90}$ | 271.38 | -54.74 | No
Leo IV |  154.0 $\pm$ 5.0 | $16.40_{-1.15}^{+1.01}$ | 265.46 | 56.51 | Yes
Leo V |  173.0 $\pm$ 5.0 | $17.65_{-1.03}^{+0.91}$ | 261.86 | 58.53 | No
Pegasus III |  215.0 $\pm$ 12.0 | $18.30_{-0.97}^{+0.89}$ | 69.85 | -41.83 | No
Pisces II |  183.0 $\pm$ 15.0 | $17.30_{-1.09}^{+1.00}$ | 79.21 | -47.11 | No
Reticulum II B |  32.0 $\pm$ 2.0 | $18.88_{-0.37}^{+0.39}$ | 266.30 | -49.74 | No
Segue 1 |  23.0 $\pm$ 2.0 | $19.12_{-0.58}^{+0.49}$ | 220.48 | 50.41 | Yes
Tucana II B |  57.5 $\pm$ 5.3 | $19.02_{-0.53}^{+0.58}$ | 328.09 | -52.32 | No
Ursa Major I |  97.3 $\pm$ 5.85 | $18.26_{-0.27}^{+0.29}$ | 159.36 | 54.43 | No
Ursa Major II |  34.7 $\pm$ 2.1 | $19.44_{-0.39}^{+0.41}$ | 152.46 | 37.44 | Yes
Willman 1 |  38.0 $\pm$ 7.0 | $19.53_{-0.50}^{+0.50}$ | 158.57 | 56.78 | Yes
Cetus |  780.0 $\pm$ 40.0 | $16.28_{-0.19}^{+0.20}$ | 156.47 | -78.53 | No
Eridanus II |  366.0 $\pm$ 17.0 | $17.28_{-0.31}^{+0.34}$ | 249.78 | -51.64 | No
Leo T |  407.0 $\pm$ 38.0 | $17.49_{-0.45}^{+0.49}$ | 214.85 | 43.66 | No
Table 1: The dwarf galaxies considered in this analysis. Note that we have
excluded Fornax from our main analysis (due to it containing globular clusters
[26]). The $J$-factor estimate are from Ref. [27] and are given in units of
GeV2/cm5.

In this paper, we evaluate the sensitivity of the proposed Advanced Particle-
astrophysics Telescope (APT) [28, 29] to dark matter annihilating in the dwarf
galaxies of the Milky Way.111The proposed APT [28, 29] is planned to follow
the ADAPT (Antarctic Demonstrator for the Advanced Particle-astrophysics
Telescope) mission, which is scheduled for a 30 day, sub-orbital flight in
2025. For more details, see https://adapt.physics.wustl.edu/. To this end, we
perform a stacked analysis of simulated data from the directions of 30 Milky
Way dwarf galaxies, assessing the constraints on annihilating dark matter that
could be achieved by such an instrument. We then estimate the projected
sensitivity of APT to dark matter in a scenario motivated by the Galactic
Center Gamma-Ray Excess. If the Galactic Center Excess is generated by
annihilating dark matter, we find that APT will be able to detect gamma-ray
signals from several dwarf galaxies at high significance. We conclude that
such a telescope would be able to definitively confirm or rule out dark matter
interpretations of the long-standing Galactic Center Gamma-Ray Excess.

##  II Dark Matter Annihilation in Milky Way Dwarf Galaxies

Figure 1: A comparison of the constraints on the dark matter annihilation
cross section (to $b\bar{b}$) attained from four years of simulated (left) or
real (right) Fermi data, from the directions of 15 dwarf galaxies. The solid
lines and the surrounding green and yellow bands denote the median constraint
and the range of constraints attained in 68% and 95% of the simulated
realizations, respectively. In the right frame, we show the results from Ref.
[30], which were derived from four years of real Fermi data. The similarity
between the simulated and real constraints demonstrates that our model
provides an adequate description of the relevant backgrounds.

The gamma-ray signal from annihilating dark matter can be calculated as
follows:

$\displaystyle\frac{dN_{\gamma}}{dE_{\gamma}}$
$\displaystyle=\frac{\langle\sigma v\rangle}{8\pi
m_{X}^{2}}\,\frac{dN_{\gamma}}{dE_{\gamma}}\bigg{|}_{\rm
ann}\int_{\Delta\Omega}\int_{\rm los}\rho_{X}^{2}\,dl\,d\Omega$ (1)
$\displaystyle=\frac{\langle\sigma v\rangle}{8\pi
m_{X}^{2}}\,\frac{dN_{\gamma}}{dE_{\gamma}}\bigg{|}_{\rm
ann}\,J(\Delta\Omega),$

where $\langle\sigma v\rangle$ is the thermally-averaged dark matter
annihilation cross section, $m_{X}$ is the mass of the dark matter particle,
and $dN_{\gamma}/dE_{\gamma}|_{\rm ann}$ is the spectrum of gamma rays
produced per annihilation. The $J$-factor, $J(\Delta\Omega)$, is defined as
the square of the dark matter density, $\rho_{X}$, integrated over a solid
angle, $\Delta\Omega$, and along the line-of-sight, $l$. The dark matter
distributions of the Milky Way dwarf galaxies (and their corresponding
$J$-factors) are constrained by spectroscopic measurements of stellar
velocities. In our main analysis, we have used the $J$-factors provided by
Pace and Strigari in Ref. [27].

Dwarf galaxies are attractive targets for dark matter searches due to their
low astrophysical backgrounds. As gamma-ray telescopes become larger and more
capable of detecting fainter sources, however, even modest backgrounds will
become increasingly important. In our analysis, we have adopted a background
model that consists of three components: 1) the gamma rays associated with
unresolved point sources, 2) the background of isotropically distributed gamma
rays, and 3) the diffuse emission associated with cosmic ray interactions in
the interstellar medium. Note that these backgrounds do not originate from the
dwarf galaxies themselves, but rather are associated with emission that is
coincidentally produced along the lines-of-sight to these systems. Although
millisecond pulsars in dwarf galaxies could potentially represent another
background for dark matter searches, the gamma-ray fluxes arising from such
objects are not expected to significantly impact such efforts [31]. Possible
exceptions are the Fornax and Sagittarius dwarf galaxies, which are known to
contain globular clusters [26, 32], making it more likely that they harbor a
significant population of millisecond pulsars. To be conservative, we have not
included Fornax or Sagittarius in our main analysis.

Far away from the Galactic Plane, the gamma-ray emission from unresolved point
sources is dominated by blazars, and we take the source count distribution of
this population to follow the triply-broken power-law model described in Ref.
[33]. For each simulated observation of a dwarf galaxy, we draw from this
distribution (up to sources as bright as $S\sim 10^{-6}\,{\rm ph}\,{\rm
cm}^{-2}\,{\rm s}^{-1}$ and extrapolated down to $S\sim 10^{-13}$, where $S$
is the photon flux integrated above 0.1 GeV) to determine whether any such
sources are present in that direction and, if so, their fluxes. We take the
spectral shape of each blazar to be the same as that of the measured
extragalactic gamma-ray background [34]. Note this component of the gamma-ray
background is fundamentally non-Poissonian in nature. The integral of this
distribution constitutes approximately 60% of the total extragalactic gamma-
ray background. We take the remaining 40% of this background to be isotropic,
arising from diffuse mechanisms or from sources that produce no more than one
photon in the data sets we will consider here. For the Galactic diffuse
emission, we adopt the model $glliemv05.fit$, as provided in Ref. [30]. Unlike
contributions from unresolved blazars and the isotropic background, the
spectrum and intensity of the Galactic diffuse emission depends on the
location of a given dwarf galaxy on the sky.

Figure 2: The projected constraints on the dark matter annihilation cross
section (to $b\bar{b}$ or $\tau^{+}\tau^{-}$) for 10 years of APT data from
the directions of 30 Milky Way dwarf galaxies (see Table 1). The solid lines
and the surrounding green and yellow bands denote the median constraint and
the range of constraints attained in 68% and 95% of the simulated
realizations, respectively. The dashed curve is the annihilation cross section
predicted for a dark matter candidate that is a (velocity-independent) thermal
relic [20], while the dot-dashed line is the current constraint from Fermi
data, as presented in Ref. [35]. Figure 3: As in Fig. 2 but for other
annihilation channels. These projected constraints are based on 10 years of
simulated APT data from the directions of 30 Milky Way dwarf galaxies (see
Table 1). The solid lines and the surrounding green and yellow bands denote
the median constraint and the range of constraints attained in 68% and 95% of
the simulated realizations, respectively. The dashed curve is the annihilation
cross section predicted for a dark matter candidate that is a (velocity-
independent) thermal relic [20].

Once we have determined the total gamma-ray flux from the direction of a given
dwarf galaxy, we integrate over each energy bin and multiply by the
instrumental exposure (using the acceptance evaluated at the average energy of
that bin) to determine the mean number of photons that are observed in each
energy bin and from within $0.5^{\circ}$ of the dwarf galaxy’s center. We then
draw from a Poisson distribution in each bin to determine the number of
photons that are observed in that realization, and use this information to
calculate the likelihood of attaining the simulated data as a function of
$\langle\sigma v\rangle$ (for a given value of the dark matter mass and
annihilation channel). Furthermore, for each simulated dwarf, we draw the
value of $\log_{10}(J)$ from a Gaussian distribution with a central value and
width equal to the quoted best-fit value and uncertainty [27]. In evaluating
the stacked likelihood, we follow the approach described in Ref. [30]. We
repeat this procedure 1000 times for each dark matter mass, cross section, and
channel in order to attain the resulting statistical distribution.

To assess the adequacy of our simulation and background model, we simulated
four years of Fermi data from the directions of 15 dwarf galaxies and compared
the resulting constraints to those attained by the Fermi Collaboration from
four years of real data [30]. In performing this comparison, we adopted the
same $J$-factors (and their uncertainties) as in Ref. [30], and used the
energy-dependent acceptance as given in Ref. [36]. For each realization, we
calcross sectionculated the 95% confidence-level upper limit on the
annihilation cross section, corresponding to a change in the log-likelihood
(relative to $\langle\sigma v\rangle=0$) of $2\Delta\ln\mathcal{L}=-3.84$. As
can be seen in Fig. 1, our simulated constraints are very similar to those
found using real Fermi data, demonstrating that our model provides a good
description of the backgrounds relevant to such an analysis.

##  III The Projected Sensitivity of APT to Annihilating Dark Matter

To assess the projected sensitivity of the Advanced Particle-astrophysics
Telescope (APT) [28, 29] to dark matter particles annihilating in Milky Way
dwarf galaxies, we simulated 10 years of APT data, adopting an acceptance as
given in Ref. [28], considering the 30 dwarf galaxies listed in Table 1
(exempting Fornax), and using the $J$-factor determinations from Ref. [27].

We show the results of this exercise in Fig. 2, for the cases of annihilation
to $b\bar{b}$ or $\tau^{+}\tau^{-}$. Due to the much larger acceptance of APT,
these projected constraints are significantly more stringent than those
derived from Fermi data [4, 22]. For comparison, we also include in these
frames the annihilation cross section for a dark matter candidate that is a
(velocity-independent) thermal relic [20], as well as the latest constraints
derived from Fermi data [35]. The constraints projected for other annihilation
channels are shown in Fig. 3.

##  IV Testing the Origin of the Galactic Center Gamma-Ray Excess

Figure 4: The projected ability of APT (with 10 years of data) to measure the
dark matter mass and annihilation cross section in a scenario with
$m_{X}=45\,{\rm GeV}$ and $\langle\sigma v\rangle=2\times 10^{-26}\,{\rm
cm}^{3}/{\rm s}$ (to $b\bar{b}$), as motivated by the Galactic Center Gamma-
Ray Excess. The star and surrounding contours represent the best-fit value and
the 1, 3 and 5$\sigma$ confidence intervals, respectively. In such a scenario,
we project that APT could exclude the null hypothesis at a level of
$2\Delta\ln\mathcal{L}\approx-200$, corresponding to a significance of
14$\sigma$. Figure 5: The projected ability of APT (with 10 years of data) to
measure the gamma-ray fluxes (integrated above 0.1 GeV) from individual dwarf
galaxies in a scenario with $m_{X}=45\,{\rm GeV}$ and $\langle\sigma
v\rangle=2\times 10^{-26}\,{\rm cm}^{3}/{\rm s}$ (to $b\bar{b}$), as motivated
by the Galactic Center Gamma-Ray Excess. These fluxes are compared to the
$J$-factors of the dwarfs, as integrated within a radius of $0.5^{\circ}$.
These results were attained in a single (but representative) realization of
our simulation, showing each dwarf that was detected with a significance of
$2\sigma$ or higher. Such a data set would allow us to test whether the gamma-
ray fluxes from dwarf galaxies are proportional to the corresponding
$J$-factors, providing an unambiguous test of dark matter interpretations of
the Galactic Center Gamma-Ray Excess.

In the previous section, we carried out our simulations under the assumption
that there is no signal from annihilating dark mater, and derived the
constraints that could be attained by an instrument such as APT. It is
possible, however, that such a signal could be found in the data, in
particular in light of the long-standing Galactic Center Gamma-Ray Excess [9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. Motivated by this excess, we consider
in this section a scenario in which the Galactic Center excess is generated by
annihilating dark matter, evaluating the sensitivity of APT to a dark matter
candidate that is capable of generating this signal.

For concreteness, we will consider a dark matter particle with a mass of
$m_{X}=45\,{\rm GeV}$ and that annihilates to $b\bar{b}$ with a cross section
of $\langle\sigma v\rangle=2\times 10^{-26}\,{\rm cm}^{3}/{\rm s}$. We again
perform a simulation of 10 years of APT data from the directions of 30 dwarf
galaxies, but calculate the change in the log-likelihood relative to the best
fit value of $m_{X}$ and $\langle\sigma v\rangle$. The results of this
exercise are shown in Fig. 4. Relative to the best-fit parameter values,
$\langle\sigma v\rangle=0$ is disfavored in this scenario at a level of
$2\Delta\ln\mathcal{L}\approx-200$, ruling out the null hypothesis with a
significance of approximately 14$\sigma$.

The gamma-ray signal from dark matter annihilating in a given dwarf galaxy is
proportional to that galaxy’s $J$-factor, providing us with a powerful way to
distinguish dark matter annihilation products from astrophysical backgrounds
(which would not be expected to scale with $J$). In a scenario with
$m_{X}=45\,{\rm GeV}$ and $\langle\sigma v\rangle=2\times 10^{-26}\,{\rm
cm}^{3}/{\rm s}$ (to $b\bar{b}$), we find (in our median simulation) that APT
will detect 7 dwarf galaxies at $>2\sigma$ significance, 4 at $>3\sigma$, and
3 at $>5\sigma$. This data would allow us to test the whether the the gamma-
ray fluxes from these galaxies are, in fact, proportional to the corresponding
$J$-factors. In Fig. 5, we show the results of a representative realization of
our simulation, showing the gamma-ray fluxes from each of the 7 dwarfs that
were detected with greater than $2\sigma$ significance.

##  V Summary and Conclusions

Gamma-ray observations of dwarf galaxies can be used to place stringent
constraints on annihilating dark matter. At present, such searches are
statistically limited, and thus would significantly benefit from experiments
capable of detecting more gamma-ray photons from such systems. In this
context, we have evaluated in this paper the sensitivity of a future, large-
acceptance, space-based gamma-ray telescope, focusing on the case of the
proposed Advanced Particle-astrophysics Telescope (APT).

As shown in Figs. 2 and 3, we project that an APT-like telescope would be very
sensitive to annihilating dark matter particles, probing annihilating cross
sections associated with thermal relics for masses up to $m_{X}\sim 600\,{\rm
GeV}$ (for the case of annihilation to $b\bar{b}$). In contrast, Fermi is
currently only sensitive to such particles if they are lighter than $m_{X}\sim
50\,{\rm GeV}$ [4, 35].

If the Galactic Center Gamma-Ray Excess is generated by annihilating dark
matter, the corresponding signal from dwarf galaxies would be unambiguously
detected by APT. In such a scenario, we find that APT would detect 7 dwarf
galaxies with a significance of at least 2$\sigma$, and 3 dwarfs with a
significance of 5$\sigma$ or greater (in our median simulation). From such
measurements, it could be established whether the gamma-ray fluxes from dwarf
galaxies are proportional to the corresponding $J$-factors, providing us with
a smoking gun signature of annihilating dark matter.

In our simulations, we have considered 30 known dwarf galaxies and have used
currently available determinations of their $J$-factors. It is anticipated,
however, that many new dwarf galaxies will be discovered in the Rubin/LSST
era, increasing the sensitivity of gamma-ray searches for dark matter
annihilation products [37, 38, 39, 40]. Furthermore, spectroscopic
measurements of stellar velocities in dwarf galaxies will continue to improve
our ability to determine the $J$-factors of these systems, further improving
the sensitivity of stacked dwarf analyses of gamma-ray data. For these
reasons, the actual sensitivity of APT to annihilating dark matter could
plausibly exceed the projections that we have presented in this study.

###### Acknowledgements.

We would like to thank Alex Drlica-Wagner for helpful discussions. FX would
further like to thank Rich Kron for his ongoing support. This work is
supported by the Fermi Research Alliance, LLC under Contract No. DE-
AC02-07CH11359 with the U.S. Department of Energy.

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