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 $Ax=b$.

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Correspondence of forms of the same solution of a system of linear equations [migrated]

I am currently reading Mathematics for Machine Learning book (p.27-28, freely available). And I am really confused with how two different forms of a solution of a system of linear equations correspond ...
Egor's user avatar
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4 votes
2 answers
125 views

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} \...
Tardis's user avatar
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1 vote
1 answer
333 views

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, ...
Juan's user avatar
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6 votes
1 answer
184 views

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 ...
Gabi's user avatar
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2 votes
0 answers
250 views

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 ...
user369335's user avatar
0 votes
0 answers
119 views

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, ...
mukhujje's user avatar
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3 votes
1 answer
143 views

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 \...
Arnaud Casteigts's user avatar
1 vote
1 answer
238 views

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 ...
Danish's user avatar
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1 vote
1 answer
152 views

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}...
RedOct's user avatar
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6 votes
2 answers
633 views

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 ...
Chris H's user avatar
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2 votes
1 answer
119 views

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 $m \times 1$ ...
Leon Staresinic's user avatar
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0 answers
143 views

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 &...
hopeless's user avatar
0 votes
1 answer
101 views

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 ...
user8469759's user avatar
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0 answers
37 views

Does this recurent matrix sequence admit an explicit writing?

I have sequence defined by : 𝐏(n+1)=(𝐈−(Ф.𝐏(n).Ф′+𝐐).𝐇′.(𝐇.(Ф.𝐏(n).Ф′+𝐐).𝐇′+𝐑)^(−𝟏).𝐇).(Ф.𝐏(n).Ф′ +𝐐) Where : P(n), Q, R are square, NxN, symmetric, positive semidefinite. R is square, ...
Poxknot's user avatar
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0 answers
70 views

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 ...
Kanghun Kim's user avatar
3 votes
1 answer
517 views

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 ...
Kanghun Kim's user avatar
1 vote
1 answer
157 views

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 ...
curiouser's user avatar
3 votes
0 answers
138 views

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 ...
André Schlichting's user avatar
7 votes
1 answer
474 views

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, ...
Kanghun Kim's user avatar
8 votes
7 answers
1k views

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{...
Subhankar Ghosal's user avatar
1 vote
0 answers
63 views

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 ...
JannyBunny's user avatar
2 votes
0 answers
56 views

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 ...
Cup Y's user avatar
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9 votes
1 answer
600 views

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 $J$ ...
user369335's user avatar
1 vote
1 answer
116 views

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 ...
charmin's user avatar
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1 vote
0 answers
28 views

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, ...
Kyle Shapiro's user avatar
-3 votes
1 answer
482 views

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 ...
GKR's user avatar
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1 vote
1 answer
229 views

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} $\Sigma$ & $\Gamma$ ...
Bayesian's user avatar
2 votes
0 answers
97 views

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 $, ...
newbie's user avatar
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2 votes
2 answers
157 views

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 ...
user36313's user avatar
  • 123
1 vote
0 answers
196 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 ...
gtg's user avatar
  • 11
3 votes
2 answers
633 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 ...
tomate's user avatar
  • 503
3 votes
1 answer
325 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. ...
StatsyBanksy's user avatar
1 vote
0 answers
133 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 ...
Hvjurthuk's user avatar
  • 573
3 votes
2 answers
338 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 ...
Wijdan Mt's user avatar
0 votes
1 answer
136 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 ...
Apprentice's user avatar
1 vote
0 answers
84 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 ...
Mich55's user avatar
  • 11
1 vote
2 answers
336 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 ...
user46230's user avatar
  • 268
10 votes
3 answers
566 views

System of quadratic equations

Let $B_1,\ldots,B_s$ be $(s\times s)$ symmetric real matrices and $x=\left(x_1,\ldots,x_s\right)^\prime$ a $(s\times 1)$ vector of unknowns. Is there a way or reference theory for studying the ...
meie73's user avatar
  • 121
1 vote
0 answers
151 views

Optimization problem on trace of complex matrix product

Given a complex rectangular matrix $A$ $(k \times n)$, I am interested in solving the following optimization problem over $(k\times n)$ complex matrices $x$: $$ \mathrm{arg}\max_X \,\mathrm{trace}(X^...
hichem hb's user avatar
  • 377
1 vote
2 answers
269 views

Proving that a system of polynomial matrix equations over $\mathbb{F_2}$ has no solution

I am working on a problem involving nilpotent matrices over $\mathbb{F}_2$ and I was able to reduce it to proving that the system \begin{equation} \begin{cases} A^2+ BC+ BCA+ ABC+A = I_4 \\ ...
Mashed Potato's user avatar
3 votes
1 answer
313 views

Is the solution to the Fredholm integral equation of the first kind a continuous analogue of Cramer's rule for matrix equations?

When we have a system of of $n$ linear equations represented by $$A \vec{x} = \vec{b} $$ with $\vec{x} = (x_{1}, x_{2}, \dots, x_{n})^{\intercal} $, we can solve for each component of this vector by ...
Max Muller's user avatar
  • 4,575
2 votes
0 answers
173 views

System of matrix equations

Problem definition: Let $x_i \in \mathbb{R}^d$ and $a_i \in [0,1]$, for all $i = 1,\dots, k$ (with $k\geq d$). Define $M(a) = \sum_{i = 1}^k a_i x_ix_i^T,$ and assume $M(a) \succ 0.$ Question: Is ...
Apprentice's user avatar
1 vote
1 answer
195 views

Matrix equation involving quadratic form

Let $X,Y\in\mathbb{R}^{n\times k}$, $\Lambda(\alpha) = \text{diag}(\alpha)$, with $\alpha\in\mathbb{R}^k$, and let $c,d\in\mathbb{R}^+$ be positive constants. Let $$A_i(\alpha) = (X\Lambda(\alpha) X^...
Apprentice's user avatar
3 votes
0 answers
47 views

Stability of matrix equation

Let $M=I+A\in \mathbb{R}^{n\times n}$ for a skew-symmetric matrix $A$ with $\|A\|<1$ in the spectral norm. Using the $LU$-decomposition of $M$, it is easy to construct a solution $L,U\in \mathbb{R}^...
user3095304's user avatar
1 vote
0 answers
173 views

What is the solution of the matrix equation $A X + X A' + B X B' + C = 0$ for $X$?

I know that the matrix equation $A X + X A' + C = 0$ is in the form of the time-continuous Lyapunov equation, so solving for $X$ is pretty trivial since the solution already and numeric solvers ...
Elias Frantz's user avatar
0 votes
1 answer
312 views

Solving two quadratic matrix equations [closed]

Given $10 \times 10$ matrices $A$ and $B$, I would like to find $10 \times 10$ matrix $X$ such that $$A = X B X^T \tag{1}$$ $$B = X A X^T \tag{2}$$ How can I solve the issue? if there is a way to ...
Moshe's user avatar
  • 11
3 votes
2 answers
218 views

A problem about determinant and matrix

Suppose $a_{0},a_{1},a_{2}\in\mathbb{Q}$, such that the following determinant is zero, i.e. $ \left |\begin{array}{cccc}\\ a_{0} &a_{1} & a_{2} \\ \\ a_{2} &a_{0}+a_{1} & a_{1}+a_{...
HilbertHunnterrrD's user avatar
0 votes
1 answer
71 views

State-dependent positive definite matrix

Consider a function $f(\mathbf{x})=\mathbf{M}_\mathbf{x}$ that outputs a nonsymmetric matrix $\mathbf{M}_\mathbf{x} \in \mathbb{R}^{N \times N}$ given an input vector $\mathbf{x} \in \mathbb{R}^N$. Is ...
Beram's user avatar
  • 1
1 vote
1 answer
357 views

How do you solve this quadratic matrix equation?

could you please help me solve this quadratic matrix equation? I look around, seems like there is no general solution for it.. $$-BX^2 + X - C = 0$$ for X, B and C are (3x3) matrices. B and C are ...
Hung's user avatar
  • 11
27 votes
1 answer
2k views

Integer matrices which are not a power

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}$In a group $G$, an element $g$ is said to be primitive if there is no $h \in G$ and integer $n >1$ such that $g = h^n$. (For clarification, I ...
Philippe Tranchida's user avatar