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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].

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60 views

Solving equations with composed constrained functions

I was lately curious about an iterative approach that would solve maths equations containing composed real functions with contraints. For example, if I have the following equation: $$ f(g(h(w))) = 0 ...
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1answer
80 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 & ...
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1answer
174 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 ...
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0answers
84 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 $A$...
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0answers
121 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 ...
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2answers
378 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 ...
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votes
3answers
437 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}^{...
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0answers
76 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{...
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1answer
47 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 ...
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1answer
109 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 ...
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2answers
123 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 ...
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vote
1answer
199 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 ...
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votes
1answer
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 ...
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0answers
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,...
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0answers
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....
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2answers
78 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 ...
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4answers
145 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
1answer
124 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....
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1answer
114 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 $...
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0answers
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 ...
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1answer
230 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 ...
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1answer
103 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
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0answers
114 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 $\mathbf{S}$...
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1answer
114 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
1answer
89 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
1answer
141 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{...
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1answer
156 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) \...
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0answers
141 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 ...
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2answers
127 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? ...
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1answer
268 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 ...
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2answers
226 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 ...
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2answers
135 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\...
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0answers
133 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 ...
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0answers
180 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
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1answer
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 $...
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1answer
90 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\...
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1answer
221 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
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1answer
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,\...
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2answers
149 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 ...
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0answers
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 \...
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1answer
854 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 ...
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0answers
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? ...
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0answers
198 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{...
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1answer
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 $\...
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votes
3answers
459 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
1answer
274 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
2answers
130 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
2answers
569 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
1answer
226 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
3answers
205 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 ...