All Questions
52 questions
15
votes
3
answers
24k
views
How to solve this quadratic matrix equation?
I would like to solve for $X$ in the matrix equation
$$
XCX + AX = I
$$
where all the matrices are $n\times n$, have real components, $X$ is positive semidefinite and $C$ is symmetric. My (possibly ...
- 395
14
votes
1
answer
2k
views
Necessary conditions for the existence of solution of Sylvester equation AX=XB
Let's consider square matrices $A_{n \times n}$, $B_{n \times n}$ and $X_{n \times n}$ with elements from $\mathbb{R}$. Could you tell me please, what would be the necessary conditions for the ...
- 335
12
votes
5
answers
9k
views
Solving Lyapunov-like equation
The following matrix equation might be a Lyapunov-like equation, but it seems hard for me to develop a simpler way to solve it. From the computation effort, I need some help for solving the special ...
- 121
11
votes
1
answer
453
views
A variant of Cholesky decomposition involving binary matrices
Studying a problem that is not directly related to linear algebra I came across the following problem.
Let $B$ be $n \times n$ symmetric matrix whose entries are non-negative integers. I would like ...
- 3,463
9
votes
2
answers
684
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 ...
- 2,712
8
votes
1
answer
2k
views
Symplectic block-diagonalization of a real symmetric Hamiltonian matrix
Given a $2n\times 2n$ real, symmetric, Hamiltonian matrix $W$ (anticommutes with the symplectic metric), is there an orthogonal, symplectic matrix $R$ such that $R^\top WR$ is block-diagonal?
Being ...
- 315
8
votes
1
answer
490
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 ...
- 7,127
7
votes
1
answer
305
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 ...
- 173
6
votes
2
answers
313
views
Representation over matrices $A_i^3=I$, $A_0A_1^\dagger+A_1A_2^\dagger+A_2A_0^\dagger=0$, $A_0^\dagger A_1+A_1^\dagger A_2+A_2^\dagger A_0=0$
I would like to know what all the possible finite-dimensional representations of the following relations are.
$$A_0^3 = A_1^3 = A_2^3 = I \tag{1}$$
$$A_0 A_1^\dagger + A_1 A_2^\dagger + A_2 A_0^\...
- 583
6
votes
3
answers
256
views
Determine unknown matrix function of particular form from known points
I encountered the following problem recently in a practical context.
Fix $n \ge 1$.
Suppose $f$ is an unknown function $\mathbb C ^ {n \times n} \to \mathbb C ^ {n \times n}$ of the form
$$ X \mapsto ...
user167785
6
votes
0
answers
450
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 ...
- 235
5
votes
1
answer
315
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
$$\begin{array}{ll} \text{minimize} & \| AX + XB - C \|_{\text{F}}^2\\ \text{subject to} & \mbox{...
- 191
4
votes
3
answers
4k
views
Non-linear matrix equation
I want to solve the following non-linear matrix equation for $X\in\mathbb{R}^{N\times N}$:
\begin{equation}
XX^{\top}+ABX^{\top}-A=0 \qquad (1)
\end{equation}
For a given matrices $A\in\mathbb{R}^{N\...
- 115
4
votes
3
answers
239
views
Properties of matrix $X=\left[\frac{1}{1-\bar\alpha_i \alpha_j}\right]_{ij}$
Let $\{\alpha_i\}_{i=1}^n$ be complex numbers such that $|\alpha_i|<1$, and consider the following $n\times n$ structured matrix
$$
X=\left[\frac{1}{1-\bar\alpha_i \alpha_j}\right]_{ij}.
$$
Such ...
- 2,712
4
votes
1
answer
1k
views
Solving the matrix equation $XX^t = A$ for binary matrix $X$
How to find all matrices $X \in \{0,1\}^{n \times m}$ that satisfy these equations?
$$X X^t = A \\ \sum_{j=1}^m x_{ij} = 2$$
These articles maybe could help us:
Completely Positive Matrices
Solving ...
- 175
4
votes
2
answers
180
views
What does the matrix-valued solution of $X = A X A^T + \operatorname{Id}$ look like?
Let $A \in \mathbb{R}^{n \times n}$ be an invertible contraction, i.e. all singular values are in $(0,1)$. By reformulating the equation
\begin{align*}
& X = A X A^T + \operatorname{Id} \tag{1}
\...
- 1,295
4
votes
3
answers
1k
views
Solving a quadratic matrix equation with fat matrix
I am trying to find an $n \times m$ fat (i.e., $m > n$) matrix $T$ that solves
$$T^T T = X$$
where $X$ is a given $m \times m$ symmetric, positive semidefinite matrix.
I saw this post, but ...
- 143
4
votes
2
answers
932
views
Steady state Kalman filter
My question is how to solve specified matrix equation (see bellow). However let me first explain background and where the equation comes from.
Kalman filter allows us to estimate state at time $t$ as ...
- 67
4
votes
1
answer
378
views
On the solvability of a matrix equation
Let $\{C_i\}_{i=1}^N$ be a set of $n\times m$ real matrices of full column-rank and such that $\mathrm{Range}[C_1,C_2,\dots,C_N]=\mathbb{R}^n$, $\{P_i\}_{i=1}^N$ a set of $m\times m$ positive definite ...
- 2,712
4
votes
2
answers
670
views
When does this linear matrix equation have a unique symmetric, positive definite solution?
I encountered the following matrix equation for $A, N, Q \in \mathbb{R}^{n \times n}$ and $A^T=-A$ and $N^T=-N$
$$[X,A]+N^TXN+Q = 0$$
where $Q$ is symmetric, positive definite. My final goal is to ...
- 41
4
votes
0
answers
262
views
Two questions about three circulant matrices
Consider the following matrix equation in $n \times n$ circulant $\pm 1$ matrices $A$, $B$, $C$
$$2AA^T+BB^T+CC^T=(4n+4)I-4J$$
where $I$ is the $n \times n$ identity matrix and $J$ is the $n×n$ matrix ...
- 696
3
votes
4
answers
359
views
Coupled Sylvester equations
Let $n \in \mathbb{N}$. Let $A,B,C$ real matrices of size $n \times n$. Let $\alpha,\beta,\gamma,\delta \in \mathbb{R}^{4}$ such that $\alpha,\beta$ are non-zero.
I an looking for two matrices $...
- 123
3
votes
2
answers
2k
views
Matrix equation with Hadamard product and its own inverse involved
I know there is an almost exactly same question here but I have further specifications. So my problem is as follows:
$$
\Omega^{-1}=\dfrac{1}{n}\left(\Omega\odot \mathbf{W}+\mathbf{X}'\mathbf{X}+\...
- 133
3
votes
1
answer
351
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
3
votes
2
answers
246
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_{...
3
votes
1
answer
155
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 \...
- 235
3
votes
2
answers
382
views
Diagonal Lyapunov equation with rank 1
Given the discrete-time Lyapunov equation (1):
$$
A^T P A - P = bb^T
$$
such that $P$ shall be diagonal and positive definite and $b$ is a column vector. How to characterize $A$ and $b$, where ...
- 909
3
votes
1
answer
277
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 ...
- 2,712
3
votes
1
answer
355
views
Conditions for a certain matrix equation to have a full rank solution
Assume that we have the following equation to solve
$$\sum_{\ell=1}^L A_\ell X_{\ell} B_{\ell} =0$$ over complex matrices
where each $A_{\ell}$ is a given $m\times n$ matrix, each $B_{\ell}$ is a ...
3
votes
0
answers
138
views
Is there a method to solve a non-linear quadratic matrix equation?
I am interested in solving the following quadratic equation: $$x^{\top} A x = \sqrt{x^{\top} B x}$$
Here, $x \in \mathbb{R^q}$ is an unknown vector, and A and B are two q$\times$q-dimensional ...
- 31
3
votes
0
answers
560
views
On a matrix algorithm involving rank-one projections
Let $\{v_i\}_{i=1}^N$ be a set of $n$-dimensional real vectors spanning $\mathbb{R}^n$. Let $p\in [0,1]$ be a rational number and consider the iteration
\begin{equation}
X_{k+1}=\frac{1}{N}\sum_{i=1}^...
- 2,712
2
votes
3
answers
324
views
Efficient algorithm for matrix equation $AXB + BXA = F$
For $n\in\mathbb{N}$, let $A\in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{n\times n}$ be two symmetric positive definite matrices and let $F\in\mathbb{R}^{n\times n}$ be arbitrary.
Is there any ...
- 205
2
votes
2
answers
185
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 ...
- 123
2
votes
0
answers
107
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 $, ...
- 61
2
votes
0
answers
176
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 ...
- 245
2
votes
0
answers
203
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 ...
- 171
2
votes
0
answers
91
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 \...
- 343
1
vote
2
answers
388
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
1
vote
1
answer
157
views
Global polynomial basis for the kernel of a matrix polynomial
Let $M(x)$ be an $m$ by $n$ matrix with entries in $\mathbb{C}[x]$. Suppose that for all $x\in \mathbb{C}$ the rank of $M(x)$ is constant and equal to $r<n$. Therefore, for any $x_0\in \mathbb{C}$ ...
- 1,488
1
vote
2
answers
485
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?
...
- 1,538
1
vote
1
answer
345
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, ...
- 61
1
vote
1
answer
270
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 ...
- 11
1
vote
1
answer
201
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
$$A_i(\alpha) = (X\Lambda(\alpha) X^...
- 245
1
vote
1
answer
333
views
On a condition for a matrix sum to be zero
Let $\{Y_i\}_{i=1}^N\in\mathbb{R}^{n\times m}$ be a set of full column rank matrices ($\mathrm{rank}(Y_i)=m$ for all $i$) and $\{P_i\}_{i=1}^N\in\mathbb{R}^{m\times m}$ be a set of positive definite ...
- 2,712
1
vote
0
answers
146
views
Identities for the determinant of a matrix similar to $\det(A)=\exp\circ\operatorname{tr}\circ\log(A)$ for different matrix functionals
The identity for the determinant of $A$ in the title is well know in matrix analysis and comes from the Jacobi's formula. I am interested in the existence of nontrivial formulas like this one (they do ...
- 573
1
vote
0
answers
192
views
What is the solution of the matrix equation $A X + X A' + B X B' + C = 0$ for $X$?
I know that the matrix equation $A X + X A' + C = 0$ is in the form of the time-continuous Lyapunov equation, so solving for $X$ is pretty trivial since the solution already and numeric solvers ...
- 11
1
vote
5
answers
639
views
Solution for $Xa + X^Tb = c$ where $X^TX = I$? [closed]
There are three known $n\times1$ vectors: $a, b, c$, along with one unknown $n\times n$ matrix: $X$. I am only interested in the $n={2,3}$ cases.
$X$ is $2\times 2$ or $3\times 3$ rotation matrix ...
- 35
1
vote
0
answers
172
views
A vanishing sum of symmetric matrices
Let $\{G_i\}_{i=1}^N\in\mathbb{R}^{n\times m}$ be a set of full column rank matrices (i.e., $\mathrm{rank}(G_i)=m$ for all $i$) and $\{P_i\}_{i=1}^N\in\mathbb{R}^{m\times m}$ be a set of positive ...
- 2,712
1
vote
0
answers
305
views
how to find all the solutions to $I+A+\cdots+A^n=0.$ [closed]
Let $GL_3(\mathbb{Z}[i])$ be the group of invertible $3\times 3$ matrices whose coefficients are Gaussian integers.I want to find all the pair $(A\in GL_3(\mathbb{Z}[i]),n\in\mathbb{Z})$ satisfying
$$...
- 197
1
vote
0
answers
296
views
Finding lower triangular matrix of an indefinite matrix
So I have the system $M = RS = RQQ^{-1}S $ and I have $R$ and $S$ currently.
I impose some constraints on $R$ in the form of $r^T$$QQ^Tr = 1$ where $r$ and $r^T$ are rows of R and their transposes. ...
- 11