All Questions
Tagged with matrices matrix-equations
111 questions
1
vote
0
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101
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Controlling the rank of a Matrix product
Let $\bf{Q}$ be an $(m-k)\times m$ matrix satisfying $\bf{QQ}^H=\bf{I}$, $\bf{W}$ an $m\times m\ell$ matrix of the form
$$\bf{}W=\left[\begin{array}{ccccc}{\bf{w}}_1 &\bf{0} &\bf{0}&\...
7
votes
2
answers
1k
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Symmetric linear least-squares solution
Given tall matrices $A$ and $Y$ and the following overdetermined linear system in square matrix $X$
$$AX=Y$$
is there an explicit formula for the least-squares solution if $X$ is constrained to be ...
4
votes
2
answers
932
views
Steady state Kalman filter
My question is how to solve specified matrix equation (see bellow). However let me first explain background and where the equation comes from.
Kalman filter allows us to estimate state at time $t$ as ...
- 5,971
0
votes
0
answers
223
views
Solving a nonlinear matrix equation
Consider the following nonlinear matrix equation:
$B=PX^{−1}AX$
where $B$ and $P$ are a $1\times n$ row vector and $A$ is a $n\times n$ matrix which are all strictly positive, and $X=diag(x_1,...,...
12
votes
5
answers
9k
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Solving Lyapunov-like equation
The following matrix equation might be a Lyapunov-like equation, but it seems hard for me to develop a simpler way to solve it. From the computation effort, I need some help for solving the special ...
- 20.2k
16
votes
2
answers
504
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The number of 0-1 normal matrices
Let $A\in\{0,1\}^{n\times n}$ be a $n\times n $ matrix with entries in the discrete set $\{0,1\}$.
My question: What is the number of matrices in $\{0,1\}^{n\times n}$ that are normal, that is, ...
- 11
2
votes
1
answer
59
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Equalities between transforms of matrices that are extremely different
I have two $2N\times 2N$ matrices, defined by blocks:
$$ A = \begin{bmatrix} a & 0 \\ 0 & 0 \end{bmatrix} $$
$$ B = \begin{bmatrix} 0 & 0 \\ 0 & b \end{bmatrix} $$
where $a$ and $b$ ...
- 21.5k
5
votes
2
answers
539
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 ...
- 52.3k
8
votes
0
answers
378
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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 ...
- 2,712
3
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0
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252
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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 ...
- 2,712
2
votes
0
answers
203
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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
0
votes
1
answer
116
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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 & ...
- 323
3
votes
0
answers
178
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$...
- 2,712
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 ...
- 1,517
9
votes
2
answers
684
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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
2
votes
1
answer
134
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 $...
- 3,741
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}.
$$
My question. Is it true ...
- 356
4
votes
3
answers
1k
views
Solving a quadratic matrix equation with fat matrix
I am trying to find an $n \times m$ fat (i.e., $m > n$) matrix $T$ that solves
$$T^T T = X$$
where $X$ is a given $m \times m$ symmetric, positive semidefinite matrix.
I saw this post, but ...
0
votes
0
answers
35
views
Solutions to this equation of the form $A(t_1,t_2)x = b(t_2)$
Given two symmetric positive definite matrices $L_1, L_2\in\mathbb{R}^{n\times n}$ and a vector $w\in\mathbb{R}^n$, under what conditions does the following system of equations of the form $M_1(t_1,...
- 403
2
votes
2
answers
254
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Low-rank solution of generalized Sylvester equation
Let $n \in \mathbb{N}$. Let $A,B,C,D$ be non-singular $n \times n$ matrices. If the matrix pencils $A-\lambda C$ and $B-\lambda D$ are regular and have disjoint spectra, then
$$AXB-CXD = 0$$
has a ...
- 20.2k
3
votes
1
answer
160
views
A matrix monotonicity question
Let $X\in\mathbb{R}^{n\times n}$ be a positive semi-definite matrix and $A\in\mathbb{R}^{n\times n}$ be a stable matrix, i.e. a matrix whose eigenvalues are strictly inside the left-half complex plane....
- 12.5k
4
votes
1
answer
286
views
Explicit formula for an LMI solution
Suppose we have a linear matrix inequality (aka LMI aka spectahedron aka linear matrix pencil):
$$A_{0}+x_{1}A_{1}+x_{2}A_{2}+\ldots+x_{m}A_{m} \succeq 0.$$
(The notation $X \succeq Y$ means that $X-...
3
votes
1
answer
287
views
Can the nonlinear matrix equation $Bx=p+\text{sgn}(x)$ have an explicit solution?
Given integers $n > 0$ and $0 \leq i \leq n$, let $D = \text{diag}(d_1, \dots, d_n)$ be positive definite, $e_i$ be the $i$th column of the $n \times n$ identity matrix, $u \in \mathbb R^n$ be such ...
1
vote
0
answers
172
views
A vanishing sum of symmetric matrices
Let $\{G_i\}_{i=1}^N\in\mathbb{R}^{n\times m}$ be a set of full column rank matrices (i.e., $\mathrm{rank}(G_i)=m$ for all $i$) and $\{P_i\}_{i=1}^N\in\mathbb{R}^{m\times m}$ be a set of positive ...
- 2,712
3
votes
1
answer
1k
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Completing the square of a matrix expression
Let $A,C\in\mathbb{R}^{m\times n}$, $n\ge m$, $B\in\mathbb{R}^{n\times m}$, and $P$ be a real positive definite $m\times m$ matrix. Denote by $\mathcal{S}^n$ the space of $n\times n$ real symmetric ...
- 20.2k
3
votes
0
answers
253
views
Solution of a quadratic matrix equation with singular coefficent
I have the following quadratic matrix equation in symmetric positive semidefinite covariance matrix $\mathbf{C}$
$$\mathbf{C}\mathbf{S}\mathbf{C}+\mathbf{C}+\mathbf{A}=\mathbf{0}$$
where $\mathbf{S}$...
2
votes
1
answer
456
views
Constrained optimization over a trace functional
Let $A\in\mathbb{R}^{n\times n}$ be a stable matrix (i.e., the eigenvalues of $A$ have negative real parts). Consider the following optimization problem in $X \in \mathbb{R}^{n \times n}$
$$\begin{...
- 1,538
2
votes
1
answer
184
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 $n \...
CommunityBot
- 1
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
$$\begin{array}{ll} \text{minimize} & \| AX + XB - C \|_{\text{F}}^2\\ \text{subject to} & \mbox{...
4
votes
1
answer
1k
views
Solving the matrix equation $XX^t = A$ for binary matrix $X$
How to find all matrices $X \in \{0,1\}^{n \times m}$ that satisfy these equations?
$$X X^t = A \\ \sum_{j=1}^m x_{ij} = 2$$
These articles maybe could help us:
Completely Positive Matrices
Solving ...
1
vote
2
answers
485
views
Closed form for integral of function of a symmetric positive definite matrix
Let $M$ be a real symmetric positive definite matrix of size $n \times n$, and let $\log M$ denote its (principal) matrix logarithm.
Is it possible to evaluate the following integral in closed form?
...
- 21.5k
8
votes
1
answer
490
views
Determinants (and traces) of linear maps of matrices
Let $k$ be a field or a commutative ring with unit and let $F:M_n(k)\to M_n(k)$ be a $k$-linear map. Suppose that $F$ is given in the form $F(X) = A_1XB_1 + \cdots + A_m X B_m$ for some $A_i,B_i\in ...
- 883
6
votes
0
answers
450
views
Can this nonlinear vector equation be solved analytically?
I have the following vector equation:
$$
{\bf Ax} + {\bf b} + {\bf Cx}^{ \circ - 1} = {\bf 0}_n
$$
Where ${\bf x}$ is a vector of unknown variables. ${\bf b},{\bf x}, {\bf x}^{\circ - 1}, {\bf 0}_n ...
- 235
3
votes
1
answer
247
views
If $A=BC$, where $A$ and $C$ are given Laplacian matrices, how to calculate $B$?
Let me be more specific: If $A=BC$, where $A$ and $C$ are given Laplacian matrices, how to calculate $B$? The graph corresponding to $A$ is a directed ring, which is strongly connected and $1_n$ and $...
- 143
4
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1
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378
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On the solvability of a matrix equation
Let $\{C_i\}_{i=1}^N$ be a set of $n\times m$ real matrices of full column-rank and such that $\mathrm{Range}[C_1,C_2,\dots,C_N]=\mathbb{R}^n$, $\{P_i\}_{i=1}^N$ a set of $m\times m$ positive definite ...
- 2,712
2
votes
1
answer
396
views
How to force least squares solution matrix to be diagonal? [closed]
I have the following matrix equation
$$AX=B$$
given $8 \times 3$ matrices $A$ and $B$. $X$ is a $3 \times 3$ diagonal matrix whose main diagonal contains the $3$ unknowns.
Whenever I solve for $X$ ...
- 43.2k
2
votes
1
answer
191
views
Reconstruct matrix given all differences of neighbors
We have an unknown $m\times n$ matrix $X=(x_{ij})_{i=1,j=1}^{m,n}$. Assume we are given measurements of the differences
$$x_{i,j+1}-x_{i,j}$$
and
$$x_{i+1,j}-x_{i,j}$$ for all $(i,j)\in \{1,\...
CommunityBot
- 1
4
votes
2
answers
671
views
When does this linear matrix equation have a unique symmetric, positive definite solution?
I encountered the following matrix equation for $A, N, Q \in \mathbb{R}^{n \times n}$ and $A^T=-A$ and $N^T=-N$
$$[X,A]+N^TXN+Q = 0$$
where $Q$ is symmetric, positive definite. My final goal is to ...
2
votes
0
answers
91
views
Matrix (geometric sum) orbit problem
Is the following algorithmic problem known to be decidable/undecidable?
Input: an element $\mathbf{v} \in \mathbb{Z}^n$, a matrix $\mathbf{A} \in GL_n(\mathbb{Z})$, and a subgroup $H \leqslant \...
- 343
3
votes
1
answer
3k
views
Solving a vector of quadratic equations
I have a system of $n \times 1$ equations
$$
0 = A\,vec(xx^t) + B x + C
$$
where
$x$ is a $n \times 1$ vector of unknowns
$x^t$ means transpose
$vec$ means $xx^t$ has been vectorized so has dimension ...
- 20.2k
1
vote
0
answers
63
views
Does the equation $\mathbf{I} = \sum_{k=1}^{m} \mathbf{U} \mathbf{B}_k \mathbf{U}^T\mathbf{A}_k$ have a special name or solution?
I have encountered the equation $\mathbf{I} = \sum_{k=1}^{m} \mathbf{X} \mathbf{B}_k \mathbf{X}^T\mathbf{A}_k$, recently.
All matrices are of dimension $n \times n$.
Is it assigned a special name?
...
- 39
24
votes
6
answers
2k
views
Cayley-Hamilton revisited
Let $(A_i)_i$ be $n\times n$ matrices with entries in a field $K$ with characteristic $0$. We consider the equation (1) $f(X)=A_kX^k+\cdots+A_1X+A_0=0_n$ where $X\in\mathcal{M}_n(K)$ is unknown. Let $...
- 12.4k
7
votes
3
answers
513
views
Trace of a nonlinear matrix equation (cont'd)
Let $X_0$ be a trace-one positive definite matrix, i.e. $X_0>0$, $\mathrm{tr}(X_0)=1$. Let $A>0$ and consider the following iteration
$$
X_{k+1} = X_k^{1/2}AX_k^{1/2},\quad k\geq 0,\quad (\star)
...
CommunityBot
- 1
7
votes
3
answers
236
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 ...
5
votes
2
answers
4k
views
Are there any algorithms for solving nonlinear matrix equations over $\mathbb{C}$?
Are there any algorithms for solving nonlinear matrix equations over $\mathbb{C}$?
I am especially interested in solving polynomial nonlinear matrix equations.
For instance, let $X$ be some matrix ...
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 ...
5
votes
1
answer
401
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
$$
X_{k+1} = X_k^{1/2}AX_k^{1/2},\quad k\geq 0,\quad (\star)
...
CommunityBot
- 1
3
votes
2
answers
2k
views
Matrix equation with Hadamard product and its own inverse involved
I know there is an almost exactly same question here but I have further specifications. So my problem is as follows:
$$
\Omega^{-1}=\dfrac{1}{n}\left(\Omega\odot \mathbf{W}+\mathbf{X}'\mathbf{X}+\...
CommunityBot
- 1
4
votes
1
answer
964
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 ...
- 53
8
votes
0
answers
196
views
Solving matrix equation $X^{-1}=\sum_{i=1}^n D_i X A_i$
Does anybody know an algorithm to solve the following matrix equation?
$$X^{-1}=\sum_{i=1}^n D_i X A_i$$
where $D_i$s are diagonal and $A_i$s are symmetric matrices.
It would be great to have an ...
- 22.1k