Equations whose unknown is a matrix, such as, for instance, algebraic Riccati equations $XAX+XB+XC+D=0$ or matrix differential equations (e.g. $\dot{X}(t)=AX(t)$. Often a subfield of [tag:na.numerical-analysis].

**9**

votes

**2**answers

355 views

### A trace-constrained maximization problem in the cone of positive definite matrices

Let $A\in\mathbb{R}^{n\times n}$ be a matrix having eigenvalues with strictly negative real part (in other words, $A$ is supposed to be Hurwitz stable). Let $\mathrm{tr}(\cdot)$ denote the trace ...

**4**

votes

**3**answers

405 views

### Non linear matrix equation

I want to solve the following non linear matrix equation for $X\in\mathbb{R}^{N\times N}$:
\begin{equation}
XX^{\top}+ABX^{\top}-A=0 \qquad (1)
\end{equation}
For a given matrices $A\in\mathbb{R}^{...

**0**

votes

**0**answers

67 views

### On the existence condition of a solution to a matrix equation

When we analysis the observability of a dynamic system, we meet a question as follows:
Let $\bar{C}=[(CA^{-i})^{T} \; (CA^{-i+1})^{T} \; (CA^{-i+2})^{T} \cdots (CA^{-i})^{T}]^{T}$ and
\begin{...

**0**

votes

**1**answer

45 views

### How to derive the solution of Tikhonov Regularization via SVD [closed]

The solution to Tikhonov Regularization is
$$x=(A^HA+\sigma^2_{min}I)^{-1}A^Hb$$
where $\sigma^2_{min}$ is the minimum of the singular values of $A$.
Then we apply $SVD$ to $A$ such that,
$$A=U\Sigma ...

**5**

votes

**1**answer

104 views

### Solving system of bilinear equations

Consider a collection of $m$ matrices $A_i$ of size $n\times n$, and a vector $b$ of size $m$. I want to solve the bilinear system $\{x^T A_i y=b_i:i=1,\dots,m\}$ in variables $x,y$. Is there an ...

**3**

votes

**1**answer

88 views

### Is there any efficient solution of the matrix equation AXB + (AXB)' + cX = D

I am trying to find the symmetric solution $X\in \mathbb{R}^{p\times p}$ of following matrix equation:
$AXB + (AXB)^T + cX = D$
where $A,B\in \mathbb{R}^{p\times p}$ are symmetric positive ...

**1**

vote

**1**answer

173 views

### On a condition for a matrix sum to be zero

Let $\{Y_i\}_{i=1}^N\in\mathbb{R}^{n\times m}$ be a set of full column rank matrices ($\mathrm{rank}(Y_i)=m$ for all $i$) and $\{P_i\}_{i=1}^N\in\mathbb{R}^{m\times m}$ be a set of positive definite ...

**2**

votes

**1**answer

176 views

### Solving a “reversed” Stein equation

Let $P$ and $Q$ be positive definite matrices. Consider the following matrix equation
$$\label{star}\tag{$\star$}
XPX^\top - P = -Q, \quad X\in\mathbb{R}^{n\times n}.
$$
My question. Is it true ...

**0**

votes

**0**answers

29 views

### Solutions to this equation of the form $A(t_1,t_2)x = b(t_2)$

Given two symmetric positive definite matrices $L_1, L_2\in\mathbb{R}^{n\times n}$ and a vector $w\in\mathbb{R}^n$, under what conditions does the following system of equations of the form $M_1(t_1,...

**1**

vote

**0**answers

30 views

### Lower bounds on eigenvalues of Lyapunov solutions

Let $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$ and let $X\in\mathbb{R}^{n\times n}$, $X=X^\top>0$ be the solution of the following Lyapunov algebraic equation
$$
AX+XA^\top=-BB^\top....

**1**

vote

**2**answers

77 views

### Low-rank solution of generalized Sylvester equation

Let $n \in \mathbb{N}$. Let $A,B,C,D$ be non-singular $n \times n$ matrices. If the matrix pencils $A-\lambda C$ and $B-\lambda D$ are regular and have disjoint spectra, then
$$AXB-CXD = 0$$
has a ...

**3**

votes

**4**answers

142 views

### Coupled Sylvester equations

Let $n \in \mathbb{N}$. Let $A,B,C$ real matrices of size $n \times n$. Let $\alpha,\beta,\gamma,\delta \in \mathbb{R}^{4}$ such that $\alpha,\beta$ are non-zero.
I an looking for two matrices $...

**3**

votes

**1**answer

120 views

### A matrix monotonicity question

Let $X\in\mathbb{R}^{n\times n}$ be a positive semi-definite matrix and $A\in\mathbb{R}^{n\times n}$ be a stable matrix, i.e. a matrix whose eigenvalues are strictly inside the left-half complex plane....

**2**

votes

**1**answer

107 views

### Quadratic matrix equation for $X\in \mathbb{C}^{ n\times p}$ with Hermitian parameters

Let $A\in \mathbb{C}^{ n\times n}$ and $B \in \mathbb{C}^{p \times p}$ be Hermitian matrices with $p < n$.
Find matrix $X$ such that $X^*AX=B.$
Solution in the case of positive definite $A$ and $...

**1**

vote

**0**answers

155 views

### A vanishing sum of symmetric matrices

Let $\{G_i\}_{i=1}^N\in\mathbb{R}^{n\times m}$ be a set of full column rank matrices (i.e., $\mathrm{rank}(G_i)=m$ for all $i$) and $\{P_i\}_{i=1}^N\in\mathbb{R}^{m\times m}$ be a set of positive ...

**3**

votes

**1**answer

188 views

### Completing the square of a matrix expression

Let $A,C\in\mathbb{R}^{m\times n}$, $n\ge m$, $B\in\mathbb{R}^{n\times m}$, and $P$ be a real positive definite $m\times m$ matrix. Denote by $\mathcal{S}^n$ the space of $n\times n$ real symmetric ...

**0**

votes

**1**answer

102 views

### A conjugation matrix $X\in \mathbb{C}^{ n\times p}$ where $p< n$

Given a Hermitian positive definite matrix $A\in \mathbb{C}^{n \times n}$ and a Hermitian matrix $B\in\mathbb{C}^{ p\times p},$ find the matrix $X$ so that $X^HAX=B$ holds where $X^H$ denotes ...

**3**

votes

**0**answers

113 views

### Solution of a quadratic matrix equation with singular coefficent

I have the following quadratic matrix equation in symmetric positive semidefinite covariance matrix $\mathbf{C}$
$$\mathbf{C}\mathbf{S}\mathbf{C}+\mathbf{C}+\mathbf{A}=\mathbf{0}$$
where $\mathbf{S}$...

**2**

votes

**1**answer

95 views

### Constrained optimization over a trace functional

Let $A\in\mathbb{R}^{n\times n}$ be a stable matrix (i.e., the eigenvalues of $A$ have negative real parts). Consider the following optimization problem in $X \in \mathbb{R}^{n \times n}$
$$\begin{...

**2**

votes

**1**answer

86 views

### Condition for non-vanishing trace

Let $A$ and $B$ be two full column rank real matrices of dimension $n \times m$, where $n \ge m$. Let $P$ be an $m\times m$ positive definite matrix.
Question: Does there always exist a symmetric $...

**5**

votes

**1**answer

140 views

### Rank-constrained least-squares solution of the Sylvester matrix equation

For the Sylvester matrix equation $AX+XB=C$, I want to find the least-squares solution $X$ via
$$\begin{array}{ll} \text{minimize} & \| AX + XB - C \|_{\text{F}}^2\\ \text{subject to} & \mbox{...

**4**

votes

**1**answer

153 views

### Least square solution to $AXB+CXD=E$

I am trying to find the least-squares solution $X$ of the following matrix equation
$$AXB+CXD=E$$
Of course, I know that this equation can be written in the form
$$(B^T \otimes A+D^T \otimes C) \...

**2**

votes

**0**answers

135 views

### Steady state Kalman filter

My question is how to solve specified matrix equation (see bellow). However let me first explain background and where the equation comes from.
Kalman filter allows us to estimate state at time $t$ as ...

**1**

vote

**2**answers

122 views

### Closed form for integral of function of a symmetric positive definite matrix

Let $M$ be a real symmetric positive definite matrix of size $n \times n$, and let $\log M$ denote its (principal) matrix logarithm.
Is it possible to evaluate the following integral in closed form?
...

**7**

votes

**1**answer

261 views

### Determinants (and traces) of linear maps of matrices

Let $k$ be a field or a commutative ring with unit and let $F:M_n(k)\to M_n(k)$ be a $k$-linear map. Suppose that $F$ is given in the form $F(X) = A_1XB_1 + \cdots + A_m X B_m$ for some $A_i,B_i\in ...

**2**

votes

**2**answers

218 views

### Standard solution to semidefinite program [closed]

I have an optimization problem of the following form
$$\text{minimize} \,\|Qa-b\|_2 \quad \text{ subject to } Q \succeq 0$$
where $a,b \in \mathbb{R}^n$ are given and the $n \times n$ square matrix ...

**0**

votes

**2**answers

133 views

### A matrix between vectors, and inequality!

I have an inequality as follows
$$s^T\phi\leq -|s|^TA$$
where $s$, $\phi$ and $A$ are vectors with appropriate dimensions. I want to prove that this inequality holds for the following too
$$s^TM\...

**2**

votes

**0**answers

122 views

### Finding positive solution for a system of linear equations using column basis

I have a binary matrix $A$ of size $m\times n$ where $n \gg m$, finding a non-negative $x$ for $Ax=b$ is not feasible, a different approach which I am currently exploring is finding solution for a ...

**6**

votes

**0**answers

174 views

### Can this nonlinear vector equation be solved analytically?

I have the following vector equation:
$$
{\bf Ax} + {\bf b} + {\bf Cx}^{ \circ - 1} = {\bf 0}_n
$$
Where ${\bf x}$ is a vector of unknown variables. ${\bf b},{\bf x}, {\bf x}^{\circ - 1}, {\bf 0}_n ...

**3**

votes

**1**answer

199 views

### If $A=BC$, where $A$ and $C$ are given Laplacian matrices, how to calculate $B$?

Let me be more specific: If $A=BC$, where $A$ and $C$ are given Laplacian matrices, how to calculate $B$? The graph corresponding to $A$ is a directed ring, which is strongly connected and $1_n$ and $...

**2**

votes

**1**answer

88 views

### Exact solution of a particular system of non-linear equations (re-formulated to matrix equation)

I have this system of $n$ non-linear equations in $n$ unknowns, arising out of my research problem. Given that $x_0=1$, I have to solve for $(x_1,x_2,\ldots,x_n).$
$$\sum_{i=0}^n x_i^2+2\sum_{j=1}^n\...

**4**

votes

**1**answer

215 views

### On the solvability of a matrix equation

Let $\{C_i\}_{i=1}^N$ be a set of $n\times m$ real matrices of full column-rank and such that $\mathrm{Range}[C_1,C_2,\dots,C_N]=\mathbb{R}^n$, $\{P_i\}_{i=1}^N$ a set of $m\times m$ positive definite ...

**3**

votes

**1**answer

156 views

### Reconstruct matrix given all differences of neighbors

We have an unknown $m\times n$ matrix $X=(x_{ij})_{i=1,j=1}^{m,n}$. Assume we are given measurements of the differences
$$x_{i,j+1}-x_{i,j}$$
and
$$x_{i+1,j}-x_{i,j}$$ for all $(i,j)\in \{1,\...

**4**

votes

**2**answers

147 views

### When does this linear matrix equation have a unique symmetric, positive definite solution?

I encountered the following matrix equation for $A, N, Q \in \mathbb{R}^{n \times n}$ and $A^T=-A$ and $N^T=-N$
$$[X,A]+N^TXN+Q = 0$$
where $Q$ is symmetric, positive definite. My final goal is to ...

**2**

votes

**0**answers

67 views

### Matrix (geometric sum) orbit problem

Is the following algorithmic problem known to be decidable/undecidable?
Input: an element $\mathbf{v} \in \mathbb{Z}^n$, a matrix $\mathbf{A} \in GL_n(\mathbb{Z})$, and a subgroup $H \leqslant \...

**3**

votes

**1**answer

811 views

### Solving a vector of quadratic equations

I have a system of $n \times 1$ equations
$$
0 = A\,vec(xx^t) + B x + C
$$
where
$x$ is a $n \times 1$ vector of unknowns
$x^t$ means transpose
$vec$ means $xx^t$ has been vectorized so has dimension ...

**1**

vote

**0**answers

56 views

### Does the equation $\mathbf{I} = \sum_{k=1}^{m} \mathbf{U} \mathbf{B}_k \mathbf{U}^T\mathbf{A}_k$ have a special name or solution?

I have encountered the equation $\mathbf{I} = \sum_{k=1}^{m} \mathbf{X} \mathbf{B}_k \mathbf{X}^T\mathbf{A}_k$, recently.
All matrices are of dimension $n \times n$.
Is it assigned a special name?
...

**3**

votes

**0**answers

196 views

### Exponential of tridiagonal matrix and boundedness of successive quotients of components

Let $a\geq 0,b>0$, $N\in \mathbb N$ and
$$ A_N = \begin{pmatrix}a(-N) & b && 0\\b & \ddots & \ddots & \\ &\ddots & \ddots & b\\0 && b & a N \end{...

**0**

votes

**1**answer

107 views

### Differential Riccati-type equation

Setup
I have recently come across an ODE of the form
$$
0 = \dot{A}(t)^TG(t) + A(t)^T\dot{G}(t) + C(t) + \lambda B(t)^TA(t)G(t) + \frac{\lambda}{2} (D(t)A(t)G(t))(D(t)A(t)G(t))^T\bar{1},
$$
where $\...

**7**

votes

**3**answers

456 views

### Trace of a nonlinear matrix equation (cont'd)

Let $X_0$ be a trace-one positive definite matrix, i.e. $X_0>0$, $\mathrm{tr}(X_0)=1$. Let $A>0$ and consider the following iteration
$$
X_{k+1} = X_k^{1/2}AX_k^{1/2},\quad k\geq 0,\quad (\star)
...

**5**

votes

**1**answer

264 views

### Trace of a nonlinear matrix equation

Let $X_0$ be a trace-one positive definite matrix, i.e. $X_0>0$, $\mathrm{tr}(X_0)=1$. Let $A>0$ and consider the following iteration
$$
X_{k+1} = X_k^{1/2}AX_k^{1/2},\quad k\geq 0,\quad (\star)
...

**3**

votes

**2**answers

124 views

### Source of equation - theorems about solving quadratic matrix equations

I have seen outlined in this comment os mathoverflow how to solve quadratic matrix equations of the form
$$
XCX + AX = I
$$
where $X \in \mathbb{R}^{n\times n}$, $C = C^T > 0 \in \mathbb{R}^{n\...

**3**

votes

**2**answers

550 views

### Matrix equation with Hadamard product and its own inverse involved

I know there is an almost exactly same question here but I have further specifications. So my problem is as follows:
$$
\Omega^{-1}=\dfrac{1}{n}\left(\Omega\odot \mathbf{W}+\mathbf{X}'\mathbf{X}+\...

**4**

votes

**1**answer

210 views

### Specific quadratic matrix equation

I am having trouble with the following matrix equation:
$(K + MU)(K + MU) = U $
$K$, $M$, and $U$ are all square matrices, the values of $K$ and $M$ are known (but they don't have a particularly ...

**7**

votes

**3**answers

203 views

### Minimizing $\|X-X_0\|_F^2+\|X^{-1}-Y_0\|^2_F$ with respect to matrix $X$

Is there an explicit solution to the problem of minimizing $$\|X-X_0\|_F^2+\|X^{-1}-Y_0\|^2_F$$ with respect to matrix $X$, where $X_0$ and $Y_0$ are given, and all matrices are real $n\times n$ and ...

**9**

votes

**0**answers

169 views

### Solving matrix equation $X^{-1}=\sum_{i=1}^n D_i X A_i$

Does anybody know an algorithm to solve the following matrix equation?
$$X^{-1}=\sum_{i=1}^n D_i X A_i$$
where $D_i$s are diagonal and $A_i$s are symmetric matrices.
It would be great to have an ...

**1**

vote

**0**answers

89 views

### Computing Moore-Penrose generalized inverse using convergent geometric series

I came across an article (published in IEEE transactions or Wiley publications) which states that moore-penrose generalized inverse can be computed using the following two equations.
Let $A$ be an $...

**9**

votes

**0**answers

353 views

### A conjecture of Blakley and Dixon about odd powers of positive matrices

In a 1966 paper Blakley and Dixon conjecture the following. Let $S$ be a symmetric matrix with nonnegative entries and let $u$ be a unit vector with nonnegative entries. For integers $k\ge j$ both odd,...

**-7**

votes

**1**answer

108 views

### Does $\|(I-A)^{-1}\|_{2} \leq 1/(1-\|A\|_{2})$ holds for matrices with spectral radius smaller then 1?

Given a symmetric positive semidefinite matrix matrix $A$, if its spectral radius $0<\rho(A)<1$, does the inequality $\|(I-A)^{-1}\|_{2} \leq 1/(1-\|A\|_{2})$ hold true?
$\|A\|_{2}$ denotes ...

**2**

votes

**1**answer

132 views

### How to force least squares solution matrix to be diagonal? [closed]

I have the following matrix equation
$$AX=B$$
given $8 \times 3$ matrices $A$ and $B$. $X$ is a $3 \times 3$ diagonal matrix whose main diagonal contains the $3$ unknowns.
Whenever I solve for $X$ ...