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Questions tagged [matrices]

Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.

3
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83 views

Sparsest similar matrix

Given a square matrix A (say with complex entries), which is the sparsest matrix which is similar to A? I guess it has to be its Jordan normal form but I am not sure. Remarks: A matrix is sparser ...
0
votes
1answer
84 views

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 & ...
3
votes
1answer
173 views

Can we classify the general linear maps such that for a fixed matrix $A \in M_n(\mathbb R)$ the conjuagation action fixes the first column?

Let $A \in GL_n(\mathbb R)$ be fixed. Let us consider the conjugation action by $G \in GL_n(\mathbb R)$, i.e., $G^{-1}AG$. I would like to see a way to identify the matrices such that the action fixes ...
4
votes
1answer
177 views

Intuitive proof of Golden-Thompson inequality

Sutter et al. [1] in their paper "Multivariate Trace Inequalities" give an intuitive proof of the following Golden-Thompson inequality: For any hermitian matrices $A,B$: $$ \text{tr}(\exp{(A+B)}) \...
6
votes
0answers
76 views

Tiling with Horn's polytopes

Let $n\ge2$ be an integer. Consider the hyperplane $H_n$ of ${\mathbb R}^n$ defined by the equation $x_1+\cdots+x_n=0$ and then the sector $P_n\subset H_n$ defined by the inequalities $x_1\le\cdots\le ...
6
votes
0answers
84 views

another extremal property of regular polygons

$\newcommand{\R}{\mathbb{R}}\newcommand{\D}[1]{\Delta_{#1}}\newcommand{\set}[1]{\{#1\}}\newcommand{\abs}[1]{\lvert#1\rvert}\newcommand{\E}{\mathbb{1}}$ In 1984 S.D.Berman, a Soviet mathematician, ...
4
votes
0answers
139 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 ...
5
votes
2answers
151 views

Generalizing Polar Decomposition of Matrices

I am trying to find a certain proof of polar decomposition of complex matrices which I think should exist more generally for a certain class of Lie groups. Recall that the polar decomposition of a ...
0
votes
0answers
12 views

Every singular DNN realization of $G$ is completely positive implies

DNN denotes doubly non-negative matrices(both entry wise non-negative and is positive semi-definite). Let $G$ be a Graph. The following two are equivalent: (a) Every non-singular DNN realization of $...
2
votes
1answer
107 views

Eigenvalues of A^T D A for positive A and diagonal D

Suppose I have a diagonal matrix $D$ whose entries are bounded in absolute value. I also have a matrix $A$ that is positive (entry-wise, so $A_{ij} > 0\ \forall\ i,j$): one can assume that the ...
7
votes
1answer
179 views

The determinant of a $4\times4$ matrix associated to some specific polynomial as follow

Let $f\in \mathbb{R}[x_1,x_2,x_3,x_4]$ defined by $$f_a(x_1,x_2,x_3,x_4)=\prod_{1\leqslant i<j\leqslant4}(x_i-x_j)^{2a_{ij}}$$ where $a=(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34})\in \mathbb{N}^6$. ...
0
votes
0answers
42 views

What can we say about the dominant Generalized eigen vector of two matrices A and B if the dominant eigen vector of A and B are known?

I have the following problem: $\bf{a} = V_{max}(A,B)$, where $V_{max}$ refers to the dominant generalized eigen vector solution and A, B are two $N \times N$ matrices (full rank). Suppose i know the ...
0
votes
0answers
114 views

Existence of an “almost” skew-symmetric matrix

Let $A\in\mathbb{R}^{3\times 3}$ be a matrix of the form $$ A=\begin{bmatrix}a_{11} & a_{12} & a_{13} \\ -a_{12} & a_{22} & a_{23} \\ -a_{13} & -a_{23} & a_{33} \end{bmatrix} $...
3
votes
1answer
198 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 ...
2
votes
0answers
87 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 $A$...
7
votes
2answers
253 views

A (linear) optimization problem subject to (linear) matrix inequality constraints

Let $A \in \mathbb{R}^{n \times n}$ be a Hurwitz matrix, i.e. $A$ satisfies $\mathrm{Re}\,\lambda_i< 0$, where $\{\lambda_i\}_{i=1}^n$ is the set of eigenvalues of $A$. Suppose that the trace of $A$...
2
votes
0answers
126 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 ...
1
vote
2answers
125 views

A “positive diagonal plus skew-symmetric” matrix decomposition

Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and strictly positive eigenvalues (note that $A$ is not required to be symmetric). My question. Do there exist an orthogonal ...
5
votes
1answer
115 views

On the linear factors of a polynomial obtained from the determinant of a matrix whose entries are related to Binomial expansion

Consider the polynomial ring $R=\mathbb C[x,y]$. Consider the matrix $A=\begin{pmatrix} x^5+y^5&5x^5&10x^5&10x^5&5x^5\\5y^5&x^5+y^5 &5x^5&10x^5&10x^5 \\10y^5&5y^5&...
2
votes
0answers
170 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 ...
4
votes
1answer
158 views

Which inner products preserve positive correlation?

Suppose we have a symmetric PD or PSD matrix M which induces an inner product $\langle \cdot, \cdot \rangle_M$. If we have that $\langle x, y \rangle > 0$ for two unit vectors $x$, $y$, are there ...
7
votes
1answer
272 views

Block matrices and their determinants

For $n\in\Bbb{N}$, define three matrices $A_n(x,y), B_n$ and $M_n$ as follows: (a) the $n\times n$ tridiagonal matrix $A_n(x,y)$ with main diagonal all $y$'s, superdiagonal all $x$'s and subdiagonal ...
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vote
0answers
88 views

Euclidean volumes under the matrix norm of one-parameter subsets of unit balls of $2 \times 2$ matrices

Prove that the Hilbert-Schmidt volume (normalized to equal 1 at $\varepsilon=1$) of the subset of the unit ball in operator norm (http://mathworld.wolfram.com/OperatorNorm.html) of the $2\times 2$ ...
3
votes
2answers
168 views

Find parameter values for which a 3x3 matrix has a triple eigenvalue

An Exceptional point generally occurs in eigenvalue problems in which the matrix is dependent on some parameter(s). The particular point in which the eigenvalues become degenerate for the parameter(s) ...
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0answers
30 views

Point-wise invertible non-linearity to reduce matrix rank [closed]

Suppose $A$ is a matrix, and its rank-$r$ SVD approximation looks like $A \approx U \Sigma V^\top$. I want to apply some invertible point-wise non-linear function $f$ and apply it to $A$ to make a new ...
2
votes
2answers
109 views

Cycle index of $(S_n \times S_n) \rtimes C_2$ acting on matrix indices by row/column permutation and transposition

Recall that there are $$\frac{n!}{\prod^n_{i = 1}i^{k_i}k_i!}$$ permutations in $S_n$ which have cycle structure $(k_1, \dots, k_n)$, that is to say they have exactly $k_1$ 1-cycles, $k_2$ 2-cycles, .....
13
votes
7answers
658 views

Is the linear span of special orthogonal matrices equal to the whole space of $N\times N$ matrices?

(Disclaimer : I know very well that $SO(N)$ has a Lie algebra of dimension $N(N-1)/2$ etc. This absolutely not the point of my question.) To make my problem more understandable, I start with the ...
2
votes
0answers
85 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 ...
1
vote
0answers
66 views

Perturbed identity in bilinear form

If I have a bilinear form $$x^TAy,$$ which I then compare to a bilinear form $$x^T(I-uu^T)^TA(I-vv^T)y,$$ where $||u||_2 = ||v||_2 = 1$, do there exist simple conditions such that the second form is ...
4
votes
1answer
62 views

Efficient Matrix Precomputation

Assume two $n \times n$ matrices $A$ and $B$ are known before hand and any precomputation can be done on them. Is there an efficient way to solve a set of linear problems: $$ \begin{split} (A+w_1 B) ...
2
votes
1answer
222 views

The minimum rank of a matrix over GF(2) when part of non-zero off-diagonal elements are set to be zero

Given an $n\times n$ matrix $A$, whose elements are over $GF\left(2\right)$ and all diagonal elements are $1$. There are $m\ (m\leq n^2-n)$ non-zero off-diagonal elements in $A$. If we are allowed to ...
6
votes
1answer
221 views

Which zero-diagonal matrices contain the all-one vector in their columns' conic hull?

Let $A$ be a non-negative zero-diagonal invertible matrix. Which $A$ make the following assertions true, which are all equivalent: The all-one vector $j$ is contained in the conic hull of $col(A)$. ...
2
votes
1answer
76 views

What can be said about the relationship between the eigenvalues of a negative definite matrix and of its Schur complement?

I have two problems related to eigenvalues of negative definite matrices: I have a matrix $M\prec0$ (symmetric and all eigenvalues are negative) and $S=M_{11}-M_{12}M_{22}^{-1}M_{21}$ by taking $M=[...
3
votes
0answers
149 views

Eigenvectors of sum of SO(3) matrices

I asked this question before on MSE but go no answers. It seems that the problem is rather difficult so I thought of trying here. Given two matrices $A,B\in SO(n)$, each describing a rotation by ...
3
votes
1answer
177 views

radical of a certain ideal of sixteen variable polynomial ring, generated by the entries of certain matrices

Consider the polynomial ring $R=\mathbb C[x_1,x_2,...,x_{16}]$, and set $$X=\begin{pmatrix} x_1 &x_2&x_3 &x_4\\ x_5&x_6& x_7&x_8\\x_9&x_{10}&x_{11}&x_{12}\\x_{13}&...
9
votes
2answers
389 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 ...
5
votes
1answer
233 views

Largest Eigenvalue of a Matrix with Special Form in terms of n

In one step of solving a difficult problem, I would like to know the largest eigenvalue of a matrix with this pattern: $$A_n = \begin{bmatrix} 0 & 0 & 0 & 0 &\dots & 0 \\ ...
4
votes
3answers
459 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}^{...
11
votes
0answers
191 views

Matrices that admit a power that is symmetric

We fix an integer $n\geq 2$. Let $S_n$ be the set of real symmetric matrices in $M_n(\mathbb{R})$. We consider the algebraic sets $Y_k=\{A\in M_n(\mathbb{R});A^k\in S_n\},k\geq 2$ and the sequence $...
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votes
0answers
54 views

Is there a transitive Lie group action on the space of matrices with rank bigger than $k$?

$\newcommand{\GL}{\operatorname{GL}}$ Let $H_{>k}$ be the space of real $d \times d$ matrices of rank bigger than $k$, for some fixed $k$. $H_{>k}$ is an open connected submanifold of $ \mathbb{...
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vote
0answers
58 views

Construction of homogeneous space

Given a Hilbertspace $\mathcal{H}$ of dimension $n<\infty$ and all hermitian matricies $Symm(\mathcal{H})$. I'd guess that the set $M_{2,2} \subset Symm(\mathcal{H})$ of all matricies of rank 4 and ...
13
votes
1answer
449 views

Tuples of 2x2-matrices simultaneously conjugate to matrices with integer entries

I am interested in the following question: Given an $n$-tuple of matrices $(A_1, \dots, A_n)\in SL(2,\mathbb R)^n$, does there exist a matrix $B\in SL(2,\mathbb R)$ such that $BA_jB^{-1}\in SL(2,\...
3
votes
0answers
105 views

An eigenvalue of certain family of matrices

Consider the matrices $$M_n=\left[\binom{i}j+\binom{2n+1-i}{j-i}+\binom{2n+1-i}j\right]_{i,j=0}^n.$$ I am convinced and hence would like to ask: Question: Is $0$ an eigenvalue of $M_n$?
2
votes
0answers
188 views

Generalization of Carleman coefficients to multivariable functions - Carleman tensor?

Recently I learned about a matrix called Carleman matrix. It is a matrix used to represent function iteration with matrix multiplying. Carleman linearization is a technique used to embed a finite ...
3
votes
0answers
78 views

Simultaneous Congruence of Two Matrices

Could you please let me know an answer to the following question on simultaneous congruence of two matrices. This question came up while trying to handle a system of PDE. QUESTION: Let $A,B\in M_n(...
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vote
0answers
71 views

Upper bounds on absolute eigenvalue of sum of two matrix

We have this iteration $$X_{k+1}=(G\cdot Jf+H)X_k+C$$ with $G$ is symmetric and nonnegative, $H$ is nonnegative. $Jf$ is the jacobian matrix of some function $f$ and we can assume it satisfy certain ...
1
vote
0answers
100 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)...
0
votes
0answers
83 views

Matrices with half columns nonnegative and half columns nonpositive real numbers

Suppose I want to study square even dimensional matrices that have the following property: Half of the columns have nonnegative entries and the other half have nonpositive entries. Now given an ...
0
votes
0answers
81 views

On the existence condition of a solution to a matrix equation

When we analysis the observability of a dynamic system, we meet a question as follows: Let $\bar{C}=[(CA^{-i})^{T} \; (CA^{-i+1})^{T} \; (CA^{-i+2})^{T} \cdots (CA^{-i})^{T}]^{T}$ and \begin{...
1
vote
1answer
104 views

Walks of odd Lengths in a Matrix

Consider the following matrix $$ A=\left[ \begin {array}{cccc} 1&1&0&0\\ 0&0&1&0\\ 0&0&1&1\\ 1&0&0&0 \end {array} \right]. $$ Assume that $B=A^k$ ...