Questions tagged [matrix-equations]
Equations whose unknown is a matrix, such as, for instance, algebraic Riccati equations $XAX+XB+CX+D=0$ or matrix differential equations (e.g. $\dot X(t)=AX(t)$). This tag is *not* meant for general systems of linear equations $Ax=b$.
200 questions
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Error characterization for $([C]_S)^{\alpha}-[C^{\alpha}]_S$ in SPD Matrices
Let $S\subset \left\{1,2,\ldots,N\right\}$ and let $[C]_{S}\in\mathbb{S}^{|S|}$ be the principal submatrix of $C$ indexed by $S$. We may view $[\cdot]_S\,:\,\mathbb{S}^{N}\rightarrow \mathbb{S}^{|S|}$ ...
- 787
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46
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What is the complexity of solving a constrained Algebraic Riccati Equation over a finite field?
Let $T=\left[\begin{array}{cc}A&B\\C&D\end{array}\right]$ be an $(m+n)\times(m+n)$ matrix over a finite field ${\mathbb F}_{q}$, where $A$ is $m\times m$ and $D$ is $n\times n$. Consider the ...
- 21
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106
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When a solution to a matrix equation can be chosen to depend smoothly in terms of the coefficients
Assume that for a given matrix $S$ there exist a positive definite, symmetric matrix $A$ such that $X=A$ is a solution to the equation $SX+XS^T=0.$ Such a solution is not unique in the space of ...
- 601
2
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1
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175
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Trace maximization for products of symmetric nonnegative matrices summing to $J_n$
Let $n$ be a positive integer and $A_1, A_2, \dots, A_k$ be a sequence of real symmetric $n \times n$ matrices with nonnegative entries, such that
$$A_1 + A_2 + \dots + A_k = J_n,$$
where $J_n$ ...
- 737
1
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0
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115
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Non-negativity of a matrix initial value problem
This is a follow-up to this. For $t \in [0,1]$, let $A(t)$ be a time-varying symmetric $n \times n$ matrix which is twice differentiable w.r.t. $t$. Let $X(t)$ be a time-varying $n \times n$ matrix ...
- 409
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0
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176
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Estimation for solution of the Lyapunov equation with semidefinite right-hand side
Consider the Lyapunov matrix equation in symmetric matrix unknown $\bf X$
$$ {\bf A}^\top {\bf X} + {\bf X} {\bf A} = − {\bf B} {\bf B}^\top$$
where the matrix $\bf A$ is Hurwitz. We know that its ...
- 21
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93
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On simplicial chain contractions compatible with local inner products
Let $K$ be a finite simplicial complex and write $C_{\bullet}(K)$ for the chain complex of the real simplicial chains on $K$. Call an inner product on the graded vector space $C_{\bullet}(K)$ local if ...
- 617
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250
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eigenvalues spectrum of random matrix expressions using determinant identities
Given a random matrix $X$ (e.g., with i.i.d. Gaussian entries) and two matrix expressions $A(X)$ and $B_\lambda(X)=B(X,\lambda)$ which satisfy (for any instance of X):
$$0=\det(\lambda I-A(X)) \iff 0=\...
- 333
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How to justify $\mathbf{S}_{x}= \frac{E_{s}}{N}I_{N\times N}$ as the best choice when the channel matrix is unknown?
I am a graduate student working in Wireless Communication, studying random matrix theory and its applications. In the context of determining channel capacity, I encountered the following generalized ...
- 161
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155
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Simultaneous left-right orbit of a set of matrices
Let $A_1,\dots, A_k$ be non-zero $n \times n$ complex matrices, and let $1 \leq r \leq n$ be an integer. I want to know if there exists a polynomial time algorithm to decide if there exist $r \times n$...
- 970
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How to solve for bounds restricting ${\Sigma}$ to symmetric-positive-semi-definiteness?
Scenario
I have a equation for a covariance matrix ${\Sigma}$ where everything but a vector of correlations is known aka $x=(x_{1}, \dots, x_{D})$ for $x_{i}\in [-1, 1]$.
Problem
I know that ${x}$ ...
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114
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Reality of connection or meromorphic function
Let's considering a family of connections: $\nabla^{\lambda}:\mathbb{C}^{*}\rightarrow \Omega^{1}(sl(2,C))$ of trivial rank2 bundle on $\mathbb{P}^{1}-\{ 0,1,\infty \}$ with simple pole. In this case, ...
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3
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Prove or disprove that the matrix equation of the form $AX+XA^{-T}=0$ has a nonsingular anti-symmetric solution $X$
I’m trying to prove that for $A=J_n(i)$, that is, the Jordan block matrix corresponding to the eigenvalue $i$ of size $n$, where $n$ is even, the matrix equation $AX+XA^{-T}=0$ has a nonsingular anti-...
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Is there a "weak" fundamental theorem of algebra for matrices?
Let $R$ be the ring of complex $n\times n$ matrices, where $n>1$.
Does every nonconstant polynomial in $R[X]$ have a root in $R$?
Note: The "strong" fundamental theorem of algebra for ...
- 398
3
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3
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491
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Least-norm solution of the matrix equation $X B + B^T X^T = A$
Let $A, B, X$ be invertible square matrices, and let $A$ additionally be symmetric. I'd like to solve the following minimization problem:
$$\underset{X}{\operatorname{argmin}} |\!| X |\!|_{{\rm F}} \...
- 463
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2
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267
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What does the matrix-valued solution of $X = A X A^T + \operatorname{Id}$ look like?
Let $A \in \mathbb{R}^{n \times n}$ be an invertible contraction, i.e. all singular values are in $(0,1)$. By reformulating the equation
\begin{align*}
& X = A X A^T + \operatorname{Id} \tag{1}
\...
- 1,389
1
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Solvability of $A X B=C$ with $X=X^\mathrm{T}$
I am studying symmetric solutions to the complex matrix equation
\begin{equation}
A X B=C,
\end{equation}
where $A$, $B$, and $C$ are $m\times n$, $n \times k$, and $m \times k$ complex matrices, ...
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490
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Efficiently solve the Sylvester equation $AX+XA = C$ where $X$ is skew-symmetric
Is there a way (more efficient than the standard vectorization) to solve the following Sylvester equation in the skew-symmetric matrix $X$ $$AX+XA = C$$ where the matrix $A$ is symmetric positive ...
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314
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Two questions about three circulant matrices
Consider the following matrix equation in $n \times n$ circulant $\pm 1$ matrices $A$, $B$, $C$
$$2AA^T+BB^T+CC^T=(4n+4)I-4J$$
where $I$ is the $n \times n$ identity matrix and $J$ is the $n×n$ matrix ...
- 418
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0
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195
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On a matrix equation with Kronecker product
Is there any work on the matrix equation in unknowns $X, Y \in {\Bbb C}^{n \times n}$ $$(X \otimes Y + Y \otimes X) \operatorname{vec}(A)=0$$ where $\otimes$ is the Kronecker product? Or, in general, ...
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206
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Does this matrix equation always have a solution?
Let $\{A_i, i\ge 3\}$ be the matrices whose columns represent numbers from $0$ to $2^i-1$ in binary form. For example,
$A_3 = \begin{bmatrix} 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \...
- 235
1
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1
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312
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Solve permutation matrix equations of the form: $X^T A X = B_1$ and $X A X^T = B_2$
I have a hard time solving the following two matrix equations for unknown permutation matrix $X \in \mathbb{R}^{n \times n}$:
$$X^T A X = B_1$$
$$X A X^T = B_2$$
where, $A$, $B_1$ and $B_2$ are all $n ...
- 11
1
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1
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224
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Symmetric linear least-squares solution ${\bf X} {\bf A} = {\bf B}$
Given the wide matrices ${\bf A} \in {\Bbb R}^{n \times m}$ and ${\bf B} \in {\Bbb R}^{p \times m} $, where $m > n > p$, form an overdetermined linear system in ${\bf X} \in {\Bbb R}^{p \times n}...
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702
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Sprinkling signs in unitary matrices
Let $A$, $B$ be $n\times n$ unitary complex matrices, such that for all indices $i,j$ we have $|a_{ij}|=|b_{ij}|$. Does there then exist diagonal unitary matrices $D,D’$ such that $DAD’=B$?
This can ...
- 1,979
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Is an almost-solvable linear equation with integer coefficients solvable?
Let $M$ be a fixed $m \times n$ rectangular matrix ($m > n$) with non-negative integer coefficients.
Does there exist a pair $(R, \epsilon)$ with the following properties:
If $b$ is a $m \times 1$ ...
- 111
1
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Constrained trace optimization with relavance to optimal asset selection
Let $D$ and $Q$ be two real $m\times m$ diagonal matrices given
$$
D=\left(\begin{array}{cccc}
d_1 & 0 & \cdots & 0\\
0 & d_2 & \cdots & 0\\
\vdots & \vdots & \ddots &...
- 43
0
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1
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229
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Least square error problem ill conditioning
I am trying to understand why I am getting an almost singular matrix in a problem I have.
The problem is a simple as
$$
\min_{X \in \mathbb{R}^{m,n}} \left\lVert AX - B \right\rVert_F^2
$$
Obvioulsy ...
- 129
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0
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104
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Follow-up question regarding real singular matrices with additional details
After my question whose answer turned out to be false, I re-examined the course of my proof, which is actually seperate from the one in my question, and found out that there's another condition, at ...
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3
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1
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Is the set of real matrices with at least one real logarithm closed under multiplication?
Let $S$ be the set of real matrices with at least one real logarithm. For some couple of its elements, for example those with at least (one real logarithm each with submultiplicative norm smaller than ...
- 296
1
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1
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257
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Solution to commutator equation in semisimple algebraic group
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Let $K$ be a field of characteristic zero, and $\SL_n$ and $\GL_n$ the special and general linear groups over $K$. Let $\Phi \in \GL_n(K), H \in ...
- 29
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0
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165
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Solvability of a matrix exponential equation - generalized matrix logarithm
For a given invertible real matrix $G\in \mathrm{GL}_d$ with $\det G>0$, we ask for a solution $B$ of the matrix exponential equation
$$
G = \exp(B) \exp\bigl(\tfrac{1}{2}(B^T-B)\bigr) .
$$
Basic ...
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Is the set of purely real square matrices, that are complex-diagonalisable, dense in the set of real matrices?
A quick search for "diagonalisable matrix" on Wikipedia immediately gives the result that the set of real-diagonalisable matrices is not dense in the set of real matrices.
I need, however, ...
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7
answers
1k
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One observation of special type of square matrix exponentiation
I was studying the following type of matrices,
$$
A = \begin{pmatrix}
1 & x_{12} & \cdots &x_{1n}\\
0 & x_{22} & \cdots &x_{2n}\\
\vdots\\
0&\cdots&0&x_{nn}
\end{...
- 271
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96
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Solve linear matrix equation involving convolution
I am facing following equation:
$$
A * X + C \cdot X = D
$$
with:
$A, C, D \in \mathbb{R}^{n \times n}$ some known matrices without any particular structure,
$X \in \mathbb{R}^{n \times n}$ the ...
- 11
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72
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any ideas on how to solve matrix equation like this $X A_i Y = B_i$
the objective function is like
$$\operatorname*{argmin}_{X,Y} = \sum_i \|X A_i Y - B_i\|_F^2$$, and $A_i$ is a diagonal matrix
I've tried gradient-descent, but as it turns out not well, I wonder if ...
- 21
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1
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735
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One question on circulant $\pm1$ matrices
Let $n > 13$ be a positive integer. Is there any $n \times n$ circulant $\pm1$ matrix $A$ satisfying the following property
$$AA^T=(n-1)I+J$$
where $I$ is the $n \times n$ identity matrix and $J$ ...
- 418
1
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1
answer
167
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Is it possible to simplify the coefficient matrix for large values of $x$?
If I have a system of $8$ linear equations for the eight variables $\{\alpha ,\beta ,\gamma ,\delta ,\eta ,\lambda ,\xi ,\rho \}$ and with the three parameters $\{x,y,z\}$ reals and $x>0$. I want ...
- 111
1
vote
0
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41
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Finding variance-minimizing weights [closed]
I'm trying to solve the following matrix calculus problem:
$\text{argmin}_{v \in R_+^K}(v'\Sigma v) \hspace{0.5pc} \text{subject to} \hspace{0.5pc} 1'v=1$
where $\Sigma$ is a well-behaved (symmetric, ...
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Multiplication of a symmetric matrices [closed]
I'm wonder if the next claim is true or not:
If A,B is a symmetric matrices over the real numbers,
and A is PSD , B is PD implies than AB is PSD.
PD - positive definite
PSD - positive semidefinite
If ...
- 1
1
vote
1
answer
279
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Two unknowns: one vector, one scalar, one equation
I would like to know if this equation is solvable for $a$ and $\alpha$:
\begin{equation}
\Sigma = \Gamma + a \left( \alpha 1^\top + 1\alpha^\top \right) +a^2 b
\end{equation}
$\Sigma$ & $\Gamma$ ...
2
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0
answers
145
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Gradient of QZ decomposition
Let $A$ and $B$ be an $m \times n$ matrix of rank $ k_1 \le \min(m,n) $ and $ k_2 \le \min(m,n) $. Then the QZ decomposition or the generalized Schur decomposition is $A = USV^T$ and $B = UTV^T $, ...
- 61
2
votes
2
answers
226
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Orthonormal solution of overdetermined linear equations
I have a two matrices $A$ and $B$ in $\mathbb{R}^{m \times n }$ ($m \gg $ n) such that there exists an orthonormal matrix $X \in \mathbb{R}^{n \times n }$, such that:
$$AX = B$$
Given that $X$ is ...
- 123
1
vote
0
answers
283
views
Solution that minimizes the sum of squared errors, with quadratic constraints
Given symmetric and positive definite $n \times n$ (real) matrices $A_1, \dots, A_m$ and $b_1, \dots, b_m \in {\Bbb R}^{n}$, I am trying to find the solution with the least sum of squared errors of ...
- 11
3
votes
2
answers
900
views
A truncated "geometric" matrix series
Let $A$ be an $n\times n$ matrix, $B$ be an $n\times m$ matrix, $C$ an $m \times m$ matrix, and consider the sum
$$\sum_{k = 0}^{N-1} A^k B C^k.$$
Is there any smart way to rewrite this sum in a way ...
- 513
3
votes
1
answer
434
views
Solution to a Sylvester equation with positive definite coefficients
Consider the following Sylvester equation, where each of the known coefficient matrices ($A$, $B$, $C$) is symmetric positive definite and has dimensions $n \times n$
\begin{align*}
C = A^TXA + B^TXB.
...
- 33
1
vote
0
answers
195
views
Identities for the determinant of a matrix similar to $\det(A)=\exp\circ\operatorname{tr}\circ\log(A)$ for different matrix functionals
The identity for the determinant of $A$ in the title is well know in matrix analysis and comes from the Jacobi's formula. I am interested in the existence of nontrivial formulas like this one (they do ...
- 583
3
votes
2
answers
407
views
Solving linear matrix equation
Given matrices $A, B, C' \in \Bbb R^{2 \times 6}$, where $'$ denotes matrix transposition, and matrix $L \in \Bbb R^{2 \times 2}$, how can one solve the following linear matrix equation in $X \in \Bbb ...
- 31
0
votes
1
answer
200
views
Product of matrices equal identity
I need to solve the following equation for the matrix $P \in\mathbb{R}^{r\times d}:$
$$
((PAP^\top)^{-1} P S P^\top (PAP^\top)^{-1})^2 = I_r,
$$
where $S$ is a symmetric $d\times d$ matrix, $A$ is a ...
- 245
1
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0
answers
101
views
In matrix product, differentiate one element with respect to another element
Background
Consider a system (roughly) along the lines of those shown in Sims, C. A. (2002). Solving linear rational expectations models, where you have
$$ AX_{t+1} = CX_t + M $$
where matrix $M$ is a ...
- 11
1
vote
2
answers
475
views
Matrix equation $P^TAP=A$
Let $A\in \mathcal{M}_{m\times m}(\mathbb R)$ , $det(A)=1$ , $A$ is positively definite. Which matrices $P$ satisfy the equation
$$P^TAP=A$$
In fact I am interested in sequences of traces $tr P^n$ of ...
- 268