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 ...

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3
votes
1answer
92 views

Upper bounds on elements of a matrix

During my research I have come across matrices this type $$C=B\left(B^T B\right)^{-1}B^T\ ,$$ where $B$ is an $m\times n$ real matrix. If $B^TB$ is not invertible, then $\left(B^T B\right)^{-1}$ ...
5
votes
0answers
144 views

Characterizing matrices with rank constraint

Given matrix $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M,b]=\{Q\in\Bbb R_{\geq0,\leq b}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad ...
4
votes
1answer
118 views

Non-negative decomposition of a non-negative matrix

Consider a matrix $A\in{\bf M}_{n\times m}({\mathbb R})$, whose entries are non-negative. Let $r$ be the rank of $A$. It is well-known that $A$ decomposes as $x_1y_1^T+\cdots+x_ry_r^T$ with ...
3
votes
1answer
97 views

A norm description for singular matrices

For $n>2$, are there norms $\parallel.\parallel_{a}$ and $\parallel.\parallel_{b}$ on $M_{n}(\mathbb{R})$ with the following property: $A\in M_{n}(\mathbb{R})$ is singular if and only if ...
2
votes
2answers
176 views

A line bundle over the manifold of singular matrices

According to answer of Denis Serre to this question, the manifold of singular matrices in $M_{n}(\mathbb{R})$ is defined as follows: $$M=\{A\in M_{n}(\mathbb{R})\mid \text{rank}(A)=n-1\}$$ So we ...
1
vote
1answer
151 views

A geometric property of singular matrices

Let $S\subset M_{n}(\mathbb{R})$ be the singular points of the equation $Det=0$. That is $S$ is the critical points of the determinant function. What matrices belongs to $S$, precisely? Let ...
1
vote
0answers
82 views

Spectrum of primitive nonnegative integer matrices

Let $P(X) = a_nX^n + \cdots + a_1X + a_0$ with $a_i \in \mathbb Z$. Question 1. Is there an efficient criterion on the $a_i$ to decide if there exists a primitive nonnegative integer matrix with ...
2
votes
2answers
152 views

Matrices with real spectrum

Assume you have a non-symmetric real square matrix of all whose eigenvalues are real. Can anything be said about it? Is it unitarily equivalent to a symmetric matrix? EDIT: Is it at least similar to ...
1
vote
0answers
106 views

Eigenvalue of product of self adjoint compact operators

Suppose A is a self adjoint $m \times m$ real matrix with eigenpairs $\{e_j, \lambda_j\}$ such that $\lambda_j > \lambda_{j + 1}$. Let $B$ be another self adjoint real $m \times m$ matrix such that ...
0
votes
0answers
111 views

A quantity associated with an algebraic variete

Let $P:\mathbb{C}^{n}\to \mathbb{C}$ be an irreducible homogenous polynomial. Is there a geometric or algebra geometric interpretation for the following quantity: The maximum number $k$ such that ...
1
vote
1answer
110 views

Matrix Submodular Inequality

Given $a,b,x > 0$ I know following the submodularity property holds: \begin{align} \frac{1}{a} - \frac{1}{a+x} \geq \frac{1}{a+b} - \frac{1}{a+b+x} \end{align} My question is, does this property ...
11
votes
0answers
251 views

Why is a matrix pencil called a pencil?

I'm trying to understand the historical context behind the word pencil in matrix pencils, or pencil of curves so on. I am aware that even Gantmacher 1959 has this terminology however I don't know ...
6
votes
1answer
241 views

Jordan decomposition of the tensor product of two matrices

I asked this question on Math.SE here, but did not get a lot of attention. I am interested in the problem of determining the Jordan decomposition of the tensor product of two unipotent matrices over ...
8
votes
1answer
137 views

Is the center of the automorphism group of a von Neumann algebra M trivial whenever M is a factor?

Question: Is the center of the automorphism group of a von Neumann algebra $\mathscr{M}$ trivial (=$\{\mathrm{id}\}$) whenever $\mathscr{M}$ is a factor (=$\mathscr{M}$ has center $\{\lambda I; ...
3
votes
1answer
222 views

Finite codimensional subvector space of $C^{*}$ algebras which contains no invertible elements

Assume that $A$ is a unital $C^{*}$ algebra. Is there a subvector space $Y\subset A$ of finite codimension which does not contain any invertible element? Let $n(A)$ be the infimum of such ...
0
votes
0answers
41 views

Orthogonalization technique after cosparse dictionary update

I'm trying to adapt the cosparse dictionary learning (DL) approach described in Analysis K-SVD to a DL method that creates the dictionary as a union of orthonormal blocks (UONB). For this I apply the ...
0
votes
0answers
43 views

Dual cone of a set of particular semidefinite cones

Let $X$ be a matrix variable $$X=\begin{pmatrix} x_1 & x_2 & x_3\\ x_2 & x_4 & x_5\\x_3 & x_5 & x_6\end{pmatrix},$$ define the cone as ...
1
vote
0answers
38 views

Bound of spectral radius of polynomial of a complex matrix

I am trying to prove or disprove the following inequality. $$ ||P(A)||_2\leq 2 \max_{\alpha\in W(A)}| P(\alpha)|,$$ where $P(\cdot)$ is a complex polynomial, $A\in \mathbb{C}^{n\times n}$ and ...
1
vote
0answers
75 views

An exact fraction of a matrix

Let $A$ be a $n \times m$ real matrix with $n<<m$ and of rank $r<n$. It is known that $A$ has exactly two distinct non-zero singular values: $\sigma_{\max}$ and $\sigma_{2}$, and also that ...
2
votes
1answer
123 views

Rank changes with matrix edits

Assume we have rank $r$ real matrix $M\in\{0,1\}^{n\times n}$ (not constrained to symmetric). Assume $W\in\{0,1\}^{n\times n}$ is rank $1$ real matrix. Case $1$: $M+W\in\{0,1\}^{n\times n}$. Could ...
0
votes
0answers
43 views

Bits of precision matrix reconstruction

We have a real rank $r$ matrix $M\in\{0,1\}^{n\times n}$. Suppose we have diagonalized using $LMR=D$. I want to recover a real matrix $\widetilde{M}$ such that maximum absolute entry of ...
0
votes
0answers
41 views

Way to parameterise sparse multi diagonal matrix

I have an NxN matrix S that looks like this: $$ S^{-1} = K^{-1} + \Lambda $$ where N is a multiple of 3, both K and S are positive definite matrices, and Lambda is $$ \Lambda = \begin{bmatrix} x ...
3
votes
0answers
89 views

Maximal compact subgroup of SL(2,|H)

The exceptional isomorphism $Spin(5,1)\simeq SL(2,\mathbb{H})$ is well-known, and I can find references that say the maximal compact of $Spin(5,1)$ is $Spin(5) \simeq Sp(2)$. So I know the answer to ...
4
votes
0answers
81 views

Can we replace 2-fold cover by n rectangles with 1-fold cover by n rectangles?

Suppose that $n$ rectangles cover every point of their union exactly twice (except for points on their boundaries). Can we partition this union into at most $n$ rectangles? I think it's pretty ...
3
votes
1answer
119 views

Is there a standard notation for off-diagonal transpose?

Given a matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, its transpose, obviously, is $A^T=\begin{pmatrix}a&c\\b&d\end{pmatrix}$. But is there a conventional way of notating the matrix ...
5
votes
1answer
426 views

Some calculus in the orthogonal group $O(n)$

How can one compute each of the following matrices, explicitly: $$\int_{O(n)} e^{g}dg$$ or $$\int_{O(n)} g^{n}dg \;\;\;\;n\in \mathbb{N} \;\;n>1$$ What is the explicite entries of the resulting ...
1
vote
0answers
187 views

A (possible) equivalent relation on the space of vector bundles

Edit: According to the essential comment of Alex Degtyarev, we revise the question as follows; Assume that $\alpha$ and $\beta$ are two oriention preserving automorphism of Lie groups $O(n)$ and ...
4
votes
2answers
207 views

An algorithm to compare two representations of a simple Lie algebra?

I have two representations of a simple (complex or real) finite-dimensional Lie algebra $S$, both given in terms of their structure constants on a given basis. the first one is the adjoint ...
3
votes
1answer
214 views

Explicit Isomorphism between $Cl(8)$ and $\mathbb{R}(16)$

I am looking for a explicit isomorphism between $Cl(8)$ (Clifford algebra over $\mathbb{R}^8$ with standard Euclidean metric) and $\mathbb{R}(16)$ (algebra of $16\times 16$ matrices over ...
1
vote
0answers
128 views

Zariski dense subgroups and conjugates

Let $H \leq \mathrm{SL}_3(\mathbb{Z})$ be a finitely generated subgroup which is not Zariski dense, and let $g \in \mathrm{SL}_3(\mathbb{Z})$. Must there be some $a \in \mathrm{SL}_3(\mathbb{Z})$ such ...
6
votes
1answer
231 views

Subadditivity of the square root for matrices

For positive numbers $a$ and $b$ we have the inequality $\sqrt{a+b} \leqslant \sqrt{a} + \sqrt{b}$. Is it true that the same holds if we take $a$ and $b$ to be positive semidefinite matrices? If not, ...
0
votes
0answers
45 views

About bi-stochastic and symmetric matrix

If you have an bi-stochastisc and symmetric matrix, what you can say about the second largest eigenvalue of this matrix? To be more precise, I would like to find upper bounds for it looking for the ...
1
vote
1answer
60 views

Are certain normal matrices circulant? (Part 2)

Let $\mathcal{F}$ denote the family of real normal matrices $A$ such that $ A^TA=\begin{pmatrix} a & b \\ b & \ddots \end{pmatrix}$, for $b>0$. As a user observed in the solution of Part 1 ...
0
votes
2answers
95 views

Simple Spectrum of Jacobi matrices

I want to call a matrix a Jacobi matrix (cause there may be different notions of Jacobi matrices) if it is a tridiagonal matrix with positive off-diagonal entries. Now, I read that the spectrum of ...
5
votes
0answers
66 views

Eigenvalues of Random Regular Bipartite Graphs

I am looking for a way of getting a good estimate of the eigenvalues of random bipartite d-regular graphs. The literature has very precise values the proofs of which are very involved and since I am ...
0
votes
1answer
143 views

Are these particular kinds of matrices well known?

Given two positive integers $n$ and $a \leq \frac{n}{2}$ consider a $n \times n$ matrix $A$ such that, all the diagonal entries are either $a$ or $a+1$ all the non-zero off-diagonal entries are ...
0
votes
2answers
98 views

About adding a negative definite rank-1 matrix to a symmetric matrix

If $B$ is a symmetric matrix then how do its eigenvalues compare to the eigenvalues of $B - vv^T$? ( where $v$ is a vector of the same dimension as $B$) I guess that the eigenvalues of $B - vv^T$ ...
1
vote
0answers
42 views

Tools to bound the singular values of a finite sum of random matrices from below?

Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...
6
votes
2answers
135 views

Is a normal matrix satisfying $A^TA=…$ circulant?

Let $A=\{a_{ij}\}$ be a normal matrix such that $a_{ij}\geq 0$ with equality iff $i=j$. Suppose that $$ A^TA=\begin{pmatrix} a & b & \cdots & b\\ b & a & \ddots & \vdots\\ ...
1
vote
1answer
76 views

QR decomposition of matrix [closed]

I have matrix $M = \begin{pmatrix} A & B \\ B^T & 0\end{pmatrix}$, where $A$ is $N\times N$, $B$ is $N\times 2$ and I have $Q$, $R$ such that $A = QR$. What is the fastest way to find $Q'$ and ...
5
votes
0answers
200 views

Existence of a matrix product from its eigenvalues

Let A and B be two positive definite, real, symmetric matrices. The eigenvalues of A, B and AB, denoted by $\lambda(X)$, obey the relation (from Bhatia): $$ \lambda^\downarrow(A) \cdot ...
1
vote
1answer
136 views

Are there good ways of relating a minor to the full determinant?

Say $A$ is a $(n-1)\times (n-1)$ matrix and we augment it by a $n^{th}$ row and a column and get a $n \times n$ matrix $B$. Is there a nice way to relate $det(B)$ and $det(A)$ and the added row and ...
-1
votes
1answer
91 views

How to show the square root function of a positive semidefinite matrix is differentiable? [closed]

How to show the square root function of a positive semidefinite matrix is differentiable? In this context PSD means symmetric PSD.
2
votes
0answers
77 views

Determinant of the sum of a psd (Kronecker) matrix and a diagonal matrix?

Let $K = K1 \otimes K2$ where $K1$ and $K2$ are positive semidefinite matrices. Let $W$ be a diagonal matrix with positive entries. (Everything is real-valued.) I want to calculate or bound $\det ...
1
vote
0answers
69 views

Bounds on smallest Eigenvalue of the Sum of a Standard Laplacian and a Diagonal Matrix

I'm trying to find upper boundaries on the smallest Eigenvalue $\lambda_1$ of $L + E$, where $L$ is a standard Laplacian of an unweighted digraph, with $\lambda_1(L) = 0$ and $E \in \{0,1\}^{n \times ...
5
votes
0answers
278 views

Transforming a binary matrix into triangular form using permutation matrices

I am interested in the complexity of the following problem: Given an $m\times n$ binary matrix $M$, can we permute its rows/columns to obtain a triangular matrix? I am also interested in ...
1
vote
1answer
120 views

The norm of a Finite Hilbert matrix

Let $H$ be an $n\times n$ Hilbert matrix, $$h_{ij}=(i+j-1)^{-1}.$$ The matrix $p$-norm corresponding to the p-norm for vectors is: $\left \| A \right \| _p = \sup \limits _{x \ne 0} \frac{\left ...
12
votes
3answers
878 views

How do we show this matrix has full rank?

I met with the following difficulty reading the paper Li, Rong Xiu "The properties of a matrix order column" (1988): Define the matrix $A=(a_{jk})_{n\times n}$, where $$a_{jk}=\begin{cases} ...
5
votes
1answer
162 views

Finding sparsest solution of a linear system

I want to find the solution with most zero-components for the following problem: $Ax=b$ for $A\in \mathbb{R}^{k\times n}, b \in \mathbb{R}^{k},k<n$, where $x$ is real and has no additional ...
1
vote
1answer
50 views

Maximizing a certain concave function over a non-convex set

I have been working on a problem which involves maximizing a concave function over a non convex constraint set. The problem is the following $$max. \frac{1}{2}x^TBx\\ s.t.\ x_i(1-x_i)=0,\ \ ...