# Questions tagged [matrix-equations]

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].

**7**

votes

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106 views

### A limiting sequence of positive definite matrices

Let $A\in\mathbb{R}^{n\times n}$ be a matrix with eigenvalues having (strictly) negative real part. Let $X\in\mathbb{R}^{n\times n}$, $X\succ 0$, be a positive definite matrix and let $P\succ 0$ be ...

**2**

votes

**0**answers

43 views

### On the existence of fixed points of a matrix iteration

Let $A\in\mathbb{R}^{n\times n}$ be a Hurwitz stable matrix (i.e. all eigenvalues of $A$ have negative real part). Let $\succeq$ denote the standard partial order in the cone of positive semidefinite ...

**0**

votes

**1**answer

84 views

### Matrix equation of the form $C A C^\intercal = D$

Consider the following square matrix
\begin{align}
A = \left(\matrix{d & 0 & -\frac12 & 0 & 0 & 0 & 0 & 0 \\
0 & d & -d+1 & -\frac12 & 0 & ...

**3**

votes

**1**answer

198 views

### On the existence of integer square root of a $3 \times 3$ positive definite matrix

As far as I know, a real square matrix $M$ has a real square root if $M$ is positive semidefinite, i.e., if all eigenvalues are nonnegative. And, in fact, its square root is unique.
I have read some ...

**2**

votes

**0**answers

87 views

### On a matrix inequality based on the Schur-Horn theorem

Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and (strictly) positive eigenvalues. (Notice that $A$ is not required to be symmetric.)
Let $A_s$ denote the symmetric part of $A$...

**2**

votes

**0**answers

125 views

### Space of change of basis matrices between two similar matrices - how to reduce it with additional tests?

Assume we have two real symmetric $n\times n$ matrices: $A, B$. We can easily test their similarity: $\textrm{Tr}(A^k)=\textrm{Tr}(B^k)$ for $k=1..n$. In this case both can be rotated to the same ...

**9**

votes

**2**answers

389 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

457 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

81 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

53 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

133 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 ...

**6**

votes

**2**answers

163 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

200 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

178 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

31 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

80 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

148 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

129 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

117 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

158 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

258 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

104 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

119 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

128 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

90 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

144 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

162 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

144 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

137 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

282 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

240 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

136 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

144 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

193 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

200 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

93 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

229 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

158 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

155 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

72 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

904 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

201 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

110 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

460 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

284 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

138 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

585 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}+\...

**3**

votes

**1**answer

254 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 ...