Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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 ...

0
votes
0answers
23 views

Generalizing expression in matrix form

I have the following expression: $\sum_{0 < s \leq S} \sum_{0 < a \leq A} (x_{a,s} - \frac{1}{A}\sum_{0 < a \leq A} x_{a,s})^2$ , which I would like to express in matrix quadratic form, as $...
0
votes
0answers
35 views

Hypermatrix and tensors

Consider a tensor defined as an element of a tensor space given by the tensor product of a number of vector spaces. This tensor is a linear combination of objects that are rank-1 tensors. Such a ...
2
votes
1answer
65 views

Eigenvalues of random matrix conditional on positive definiteness

Consider the Gaussian Orthogonal Ensemble, considered as a probability measure $\mu$ on the space of real symmetric matrices. Let $\mu|PD$ denote this measure conditioned on the event that the matrix ...
3
votes
0answers
66 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
74 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
161 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 ...
-1
votes
0answers
31 views

Condition of matrix norm proof [closed]

If $Ax = b$ $A\hat x = \hat b$ Then show that: $$\frac{1}{\operatorname{cond}(A)} \frac{\|b- \hat b\|}{\|b\|} \leq \frac{\|x- \hat x\|}{\|x\|}$$ where $\operatorname{cond}(A) = \|A\| \cdot \|A^{-...
4
votes
1answer
146 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
65 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 ...
-1
votes
0answers
52 views

Obtaining a Relation Between Positive Matrices

Consider $n \times n$ non-negative binary matrices ${\bf A}_i$ with $1\leq i \leq m$ over $\mathbb{R}$. Assume that $1\leq k \leq n$ is selected as a fixed number. Suppose that a subset of size $k$...
0
votes
0answers
67 views

How to understand the span of a matrix? [closed]

https://people.eecs.berkeley.edu/~brecht/cs294docs/week7/12.candes.recht.pdf In the section 3.3 of the paper above, the author explains the span of matrices: To apply our results to recovering low-...
6
votes
0answers
77 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, ...
2
votes
0answers
33 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
138 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
21 views

Why spectral variation bound for non-normal matrices is not a good bound for large $n$?

It is well known that if $A$ and $B$ are two $n \times n$ non-normal matrix with positive entries with largest eigenvalues $\lambda$ and $\mu$ respectively then $$|\lambda - \mu| \leq (\|A\| + \|B\|)...
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
95 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
169 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
39 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
109 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} $...
2
votes
1answer
157 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
78 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 $A$...
1
vote
2answers
130 views

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

Let $A \in \mathbb{R}^{n \times n}$ be a diagonalizable matrix with real and strictly negative eigenvalues. Suppose that the trace of $A$ is normalized to $-1$, that is $\mbox{trace}(A)=-1$. Further, ...
2
votes
0answers
99 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 ...
-2
votes
0answers
31 views

Matrix DIfferenciation [migrated]

What is $\frac{\partial{w}}{\alpha}$ for $w=(X^\top X + \alpha \boldsymbol{I})^{-1} X^\top y$ where X is an $N \times D$ matrix, y is an N dimensional vector, $\boldsymbol{I}$ is an identity matrix of ...
1
vote
2answers
103 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
112 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
166 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
133 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
226 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 ...
1
vote
0answers
85 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
160 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) ...
1
vote
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
95 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
618 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
74 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
65 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
59 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
210 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
204 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
70 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=[...
0
votes
0answers
89 views

Eigenvectors of convex combinations of stochastic matrices

Suppose we have $k$ square ($n$ by $n$) stochastic/probability matrices, $M_1, M_2,\dots ,M_k$ (so for each matrix the entries are non-negative and all column sums are one). Suppose we have a ...
3
votes
0answers
123 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
170 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
371 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
206 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
425 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
181 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 $...
2
votes
0answers
50 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{...
1
vote
0answers
57 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 ...