All Questions
59 questions
1
vote
0
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63
views
The rank of a matrix expression
I'm studying discrete-time LTI systems and state estimators for them. Recently, I studied this paper. I am facing a matrix rank calculation problem and having trouble solving it. I will provide more ...
- 4,484
0
votes
1
answer
134
views
Existence of cyclic subspace decompositions for pairs of commuting matrices
Let $\mathbb{K}$ be an arbitrary field (possibly finite). Let $V$ be a finite-dimensional vector space over $\mathbb{K}$, and let $A,B$ be two linear endomorphisms of $V$ which commute.
For $v\in V$, ...
1
vote
0
answers
49
views
Bounds on Eigenvalues After Skew-Symmetric Perturbation
Consider two matrices $\mathbf{J} \in \mathbb{R}^{n \times n}$ and $\mathbf{L} = -\mathbf{L}^T \in \mathbb{R}^{n \times n}$. I am trying to upper bound the eigenvalues of their sum:
$$\mathbf{A} = \...
- 11
5
votes
2
answers
343
views
Maximal eigenvalue of a correlation matrix with some entries fixed as zeros
Let $A$ be real a positive semidefinite matrix of dimension $n$ and with $1$s on the diagonal. Those matrices are sometimes referred to as correlation matrices. From the positivity of the minors, we ...
- 51
4
votes
1
answer
228
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A question on eigenvalue of parametric matrix
Is there a way to efficiently check if all matrices in the following set are Hurwitz stable (eigenvalues strictly in the left-hand plane)?$$\left\{ A \in \Bbb R^{n \times n} : \ell_{i,j}\leq A_{i,j} \...
- 1,216
1
vote
1
answer
1k
views
Prove that absolute value of eigenvalue is smaller than 1 [closed]
I want to prove that the absolute value of the eigenvalues of a matrix A are smaller than 1 for $$A=\left(\begin{array}{cc}
0 & -H_{11}^{-1} H_{12} \\
-H_{22}^{-1} H_{21} & 0
\end{array}\right)...
0
votes
0
answers
46
views
Lipschitz solutions to linear complementarity problems (LCP)
Let $M\in\mathbb{R}^{n\times n}$.
For $q\in\mathbb{R}^n$, define the set:
$$S_M(q)=\{y\in\mathbb{R}^n|y\ge 0,q+My\ge 0, y^\top (q+My)=0\}.$$
This is the set of solutions to the LCP $(q,M)$.
We say $...
- 183
0
votes
0
answers
193
views
Rewriting Kronecker product
im considering a pole placement problem in control theory and my controler has a specific form:
$$R=I_n\otimes q$$
where $I_n$ is the identitiy matrix of size $n$ and $q\in\Re^k$ is a vector of the ...
- 1
1
vote
2
answers
87
views
Stability of linear controller in the presence of a lag
Consider the following equation:
$\ddot{x} = -a x - b \dot{x}$
which we interpret as saying that we are trying to control $x$ by setting $\ddot{x}$.
We can rewrite this with $X = \begin{bmatrix}
...
- 427
2
votes
1
answer
536
views
Minimize matrix norm over the unitary matrices
Suppose $C_1$ and $C_2$ are some fixed $n \times n$ matrices. Define the norm $\| M \| = \sum_{i = 1}^n \max_j |M_{ij}|$. What is $\min_U \|C_1 U C_2 \|$? Here $U$ ranges over the $n \times n$ unitary ...
- 1,703
5
votes
1
answer
335
views
Projecting a symmetric matrix onto the space of linear operators with a particular eigenvalue
Specifically, I am interested in the case where one eigenvalue is exactly $0$. Given an $n \times n$ symmetric matrix, I would like to find the closest $n\times n$ symmetric matrix that has one ...
- 245
2
votes
1
answer
497
views
Does the Perron vector maximize $x^TAx$ in the simplex?
Let $\mathbf{A}$ be any $n\times n$ symmetric positive matrix ($A_{ij}>0$). It is easy to show that the solution to the following optimization problem
\begin{align}
\max_{\mathbf{x}}~~\mathbf{x^...
- 1,421
1
vote
1
answer
877
views
Minimization problem involving the inverse of an affine matrix function
I want to minimize $v^T (A+I+UQU^*)^{-1} v$, subject to $Q$ and $A$ being positive semi-definite and ${\rm trace}(Q)<1$. Here, $v$ is a given vector with unit norm, that is, $\|v\|_2=1$.
- 377
1
vote
2
answers
143
views
Controllability Gramian asymptotics for small times
Set-up. Consider the following linear controlled system
$$
\dot{y}(t) = A y(t) + B u (t), \ \ t \in [0,T], \ \ \ \ \ \ \ \ \ \ (1)
$$
where $y$ is the state of the system, $y(t) \in R ^n$, $A \in R ...
- 724
0
votes
1
answer
135
views
Solving Problem: LMIs and block matrices
I have been reading through this paper (https://ieeexplore.ieee.org/document/7995739) where I am stuck with this particular LMI. If you are familiar with control theory, the author is trying to find ...
- 103
4
votes
0
answers
249
views
Is every stable matrix orthogonally similar to a $D$-skew-symmetric matrix?
Let $\mathrm{diag}(A)$ denote the diagonal matrix with diagonal entries of $A\in\mathbb{R}^{n\times n}$ and let $\succeq$ denote the standard partial order in the cone of (symmetric) positive definite ...
- 2,712
2
votes
0
answers
248
views
A parametrization of stable matrices
Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and strictly negative eigenvalues. Furthermore, suppose that $\mathrm{tr}(A)=-1$.
My question. I'm wondering whether it is ...
- 2,712
3
votes
0
answers
243
views
An inequality concerning the solution of a Lyapunov equation
Let $A$ be an Hurwitz stable matrix (i.e. the real part of the eigenvalues of $A$ lie in the left-half plane) and $Q$ be a positive semidefinite matrix ($Q\ge 0$, for short). Let $P>0$ be the ...
- 2,712
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
1
vote
0
answers
163
views
Bounds of Procrustes problem
We denote $\|\cdot\|_F$ as the Frobenius norm of some matrix. We define $f: \mathbb{R}^{d\times r}\times\mathbb{R}^{d\times r} \rightarrow \mathbb{R}^{d\times r}$ as the following:
\begin{align}
f(A,B)...
- 11
4
votes
1
answer
289
views
A property of positive matrices
Let $\mathbb{S}^{dn} \ni X \succeq 0$ with $d,n \in \mathbb{N}$, where $X \succeq 0$ indicates that $X$ is positive semidefinite. Now partition $X$ into the block form
\begin{gather}
\begin{pmatrix}
...
- 81
1
vote
1
answer
313
views
Nonlinear low-rank approximation - corrected
I would like to state that this is related to a past question of mine which contained errors and now appears in the corrected form, with the erroneous one deleted and closed.
In my research of linear ...
- 620
0
votes
1
answer
134
views
Modification of a known optimization problem
In my research of linear algebra and optimization, I wish to modify the following well-known problem:
$ \min \lVert x-Ax \rVert$ subject to $ rank(A)\leq k $ where $ x $ is a given column vector ...
- 620
2
votes
1
answer
1k
views
Is the sum of two stable matrices also stable?
Let $A$ and $B$ be two arbitrary real matrices of the same dimension. If the eigenvalues of $A$ and $B$ are all in the left half of the complex plane, can we estimate the the location of the ...
- 23
3
votes
0
answers
298
views
Singular value decomposition of a low rank weak diagonally dominant M-matrix. When is the unitary polar matrix positive semi-definite?
Let $A$ be an $n \times n$, non-symmetric, real, weak diagonally dominant M-Matrix. Its diagonal is strictly positive, its off-diagonal is negative or zero and all its columns sum to zero. $A$ has ...
- 323
8
votes
1
answer
2k
views
Finding Toeplitz matrix nearest to a given matrix
For an arbitrary $N\times N$ Hermitian matrix $A$, I want to derive a Toeplitz matrix from $A$ such that the eigenvectors of both matrices have minimal change.
Specifically I want find the Toeplitz ...
- 495
1
vote
1
answer
606
views
The state-transition-matrix of a physical system,
Here's a simple but potential research problem that I am learning about.
Let's say I am studying a physical system that is governed by N objects. At each time, each object is either "active" and ...
- 1
1
vote
0
answers
141
views
For of a special case of $Ax>=b$, are there always integer solutions?
The input consists of two integers $n\geq 2$ and $K\geq 2$, and a vector of positive integers $b$ of size $n$. We assume that $\sum_i b_i$ is a multiple of $K$.
The output is a matrix $A(n\times n)$ ...
- 123
2
votes
0
answers
90
views
Singularities of the Quantum propagator (baby version)
Given $a,b \in \mathfrak{su}(4)$ which are taken to generate the whole algebra, consider the following map $V:\mathbb{R}^{2} \rightarrow SU(4)$:
$V : (w_1, w_{2}) \mapsto e^{(a+w_2 b)} e^{(a+w_1 b)}$
...
- 2,099
2
votes
1
answer
2k
views
Finding permutation matrix $P$ that minimizes the trace of $P C P^T D$
I have a problem that is really important for my thesis and i am not studding math so i will be very glad if you help me in this case...
thanks for your help in advance
I want to find permutation ...
- 59
3
votes
0
answers
149
views
Copositivity under tensor products
Is there any symmetric real matrix $A$ such that $A^{\otimes n}$ is copositive for all positive integers $n$, but such that A is neither positive semidefinite nor has just non-negative entries?
...
- 31
2
votes
2
answers
606
views
Solving a matrix equation $X=c \cdot AXA' +I$ with a diagonal corrections
I am now struggling to solve the matrix $X \in R^{n \times n}$ in the following equation:
$X=c \cdot AXA' - diag(c \cdot AXA')+ I$,
where
(1) $A \in R^{n \times n}$ is a given matrix whose element ...
- 241
4
votes
1
answer
538
views
Rank 1 Approximation of Elementwise Inverse Matrix
I'm wondering whether there is a good way to solve the following optimisation problem.
Given a strictly positive quadratic matrix $A$, find two diagonal matrices $D_1$ and $D_2$ so that
$$ \| D_1 A ...
- 909
2
votes
0
answers
1k
views
Incoherence of the row/column span
Due to V.Chandrasekaran., et al (p.11) : In general for any $k$-dimensional subspace of $A_{n×n}$ we have that:
$$\sqrt{(k/n)} \leq incoherence(A)\leq 1$$
where the lower bound is achieved (for ...
- 21
0
votes
1
answer
203
views
Eigenvalues of a given parametrized matrix.
Let $\mathbf{A}$ and $\mathbf{B}$ be two complex rank-one $N\times N$ positive semi-definite matrices. Let the matrix $\mathbf{C}$ be defined as
\begin{align}
\mathbf{C}=\left(\mathbf{I}*\frac{1}{\...
- 1,421
2
votes
2
answers
765
views
Why eigenvectors optimize this orthogonally constrained nonlinear minimization problem?
Given a $p \times p$ positive definite matrix $\Sigma$, why eigenvectors of $\Sigma$, stacked as columns of a matrix $R \equiv [r_1 \, r_2 \, \ldots \, r_p]$, optimize the following orthogonally ...
- 362
7
votes
3
answers
2k
views
Optimization problem on trace of rotated positive definite matrices
Given two $n \times n$ symmetric positive definite matrices $A$ and $B$, I am interested in solving the following optimization problem over $n \times n$ unitary matrices $R$:
$$
\mathrm{arg}\max_R \,\...
- 362
3
votes
1
answer
2k
views
Fast algorithm for maximizing smallest eigenvalue of linear combination of hermitian matrices
I have an engineering back ground. Due to work, I came across this problem
\begin{align}
&\max_{\lambda,y_i\in \mathbb{R}}~\lambda \\\
s.t.~&\left(\mathbf{A}_0+\sum_{i=1}^{K}y_i\mathbf{A}_i\...
- 1,421
0
votes
1
answer
2k
views
eigen-decomposition solution? is it unique?
Assume an N*N covariance matrix (Q) which is a positive definite matrix. The decoder X is assumed to be N*s, where s<=N. X is calculated to be s eigenvectors corresponding to s minimum eigenvalues. ...
- 1
3
votes
0
answers
466
views
An optimization problem over real symmetric matrices
Given an $n\times s$ matrix $P$ of positive real numbers and $T\geq n$, find (either by a formula or an algorithm) the real symmetric $n\times n$ Z-matrix $A$ which maximizes $\min\limits_{1\leq i<...
- 767
5
votes
3
answers
3k
views
A nice necessary and sufficient condition on positive semi-definiteness of a matrix with a special structure
Let
$$
A =
\begin{pmatrix}
\sum_{j\ne 1}a_{1j} & -a_{12} & \cdots & -a_{1n}\\
-a_{21} & \sum_{j\ne 2}a_{2j} & \cdots & -a_{2n}\\
\vdots & \vdots & \ddots & \...
4
votes
2
answers
359
views
A certain type of constrained Rayleigh-Ritz ratio
Let $\mathbf{A_1}$ and $\mathbf{A_2}$ be two hermitian matrices. Consider the problem
\begin{align}
\max_{\mathbf{u}^H\mathbf{u}=1}~\mathbf{u}^H\mathbf{A}_1\mathbf{u} \\\
\mathbf{u}^H\mathbf{A}_2\...
- 1,421
3
votes
0
answers
125
views
Copositivity in matrix pencils
Given two square symmetric matrices $A,B$ of the same order, the matrix pencil $P(A,B)$ is the set of linear combinations of $A$ and $B$. Finsler's theorem gives an elegant criterion for $P(A,B)$ to ...
- 6,962
-1
votes
1
answer
174
views
Regularized Gradient with respect to a matrix (with a specific structure)
Suppose we have a typical logdet function $\mathcal{L}$
$$
\mathcal{L} = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - \mathbf{q}^T(\mathbf{A}^{-1} + \mathbf{S})^{-1} \mathbf{q},
$$
where $\...
- 125
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
4
votes
2
answers
1k
views
Minimum eigenvalue of a Affine Combination of two Hermitian matrices
Consider two $N \times N$ hermitian indefinite matrices $A_1$ and $A_2$. Consider their affine combination
\begin{align}
M(t)=(1-t)A_1+tA_2
\end{align}
I am interested in the minimum eigenvalue of $M(...
- 1,421
2
votes
1
answer
299
views
Rayleight Ritz Ratio and smallest eigenvalue for a set of given matrices
I am familiar with Rayleigh Ritz Ratio for hermitian matrices. Let $A_1$ be a given $N \times N$ hermitian matrix. Then the smallest eigenvalue of $A_1$ is given by
\begin{align}
\lambda_{min}(A_1)=\...
- 1,421
5
votes
2
answers
429
views
Simultaneous maximization of two Generalized Rayleigh Ritz Ratios
Consider hermitian positive semi-definite matrices $A_1$ and $A_2$. Consider also positive definite matrices $B_1$ and $B_2$. I want to maximize the minimum of the two Generalized Rayleigh Ritz ratios ...
- 1,421
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
0
votes
1
answer
180
views
(probably simple) optimization question
Suppose you have a concave function defined over a non-polyhedral convex cone and you are interested in the infimum. What would be standard approaches to tackle the question? (The cone is actually PSD ...
- 6,962