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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4
votes
0answers
308 views

A stronger Cauchy-Schwarz inequality for traces of compression matrices

Assume that $A$ and $B$ are contractions, so $I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let $C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that: ...
0
votes
1answer
63 views

Reference request for: inverse of a non-singular M-matrix has all elements non-negative?

Does anyone know the best (earliest?) reference please for the proof that the inverse of a non-singular M-matrix has all elements non-negative?
0
votes
0answers
34 views

Nonlinear matrix equation (transpose) [duplicate]

Let $H$, $M$ and $N$ be 10 by 10 matrices over the integers. If $M$ and $N$ are known, how do you solve for $H$ from the following equation? $M = H N H^T$ where $H^T$ is the transpose of H.
2
votes
1answer
81 views

Approximation with a rank-$1$ matrix

Given a matrix $A$ (generally speaking, complex and non-square), I want to find an identically-sized matrix $D$ with ${\rm rk} D\le 1$ to minimize the induced operator norm $\|A-D\|_2$. From the ...
5
votes
7answers
477 views

Source for roots of matrix polynomials?

A matrix polynomial is a polynomial whose variables are square $n \times n$ matrices, let's say with entries in $\mathbb{C}$, and with coefficients in $\mathbb{C}$. I am seeking a source of results on ...
3
votes
0answers
130 views

Method to Generate Random Mutually Orthogonal Unitary Matrices

The title says it all, I'd like to know if such a method exists to generate random mutually orthogonal unitary matrices. If so, any supplementary references would be very much obliged. Thanks again!
8
votes
1answer
123 views

A variant of Cholesky decomposition involving binary matrices

Studying a problem that is not directly related to linear algebra I came across the following problem. Let $B$ be $n \times n$ symmetric matrix whose entries are non-negative integers. I would like ...
2
votes
0answers
142 views

Cavalieri's principle and inversion of the Vandermonde matrix

There are many examples on the Web of the use of Cavalieri's principle in determining areas and volumes of 2-D and 3-D geometrical figures. The Wikipedia link uses the principle as both a proof and ...
8
votes
2answers
474 views

Elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$

For some research work, I need to know the classification of elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$, up to conjugation. Since I essentially need $n\le 4$, I think that I can show it ...
3
votes
0answers
97 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 ...
2
votes
1answer
87 views

dimensions of strata of Pfaffian varieties

Let $V$ a complex vector space of dimension $2n$. Let us consider $W=\wedge^2V$ and the Pfaffian variety $Pf\subset \mathbb{P}W$ that parametrize degenerate skew-symmetric matrices. $Pf$ is naturally ...
0
votes
0answers
28 views

represent hankel matrix by low rank tensorial approximation

suppose that we have given following matrix \begin{matrix} x_1 & x_2 & ..x_p \\ x_2 & x_3 & ...x_{p+1} \\ . & .& . & \\ x_{N-p+1} & x_{n-p+2} &... x_n ...
0
votes
1answer
196 views

Existence of a real eigenvalue

I have a matrix $M \in \mathbb{R}^{(n+1) \times (n+1)}$ that is tridiagonal. In numerical computations I found out that I always find a real eigenvalue. My question is: Is there a theorem that ...
0
votes
0answers
107 views

A matrix rank problem over finite fields: Is that a known problem?

I have already asked the same question on cstheory.SE, but I haven't got an acceptable answer. So, I decided to ask it here. It might be a known problem, however. Let $A \odot B$ denote elementwise ...
1
vote
0answers
69 views

spectrum of a special class of tridiagonal matrices

Consider a real and symmetric tridiagonal matrix with zero diagonals and where subdiagonals and superdiagonals are equal to 1 except the (1,2)-th component being equal to $a$, i.e., ...
0
votes
1answer
95 views

Simultaneous triangularizability over a commutative ring

Let $R$ be a commutative ring with unity and $A,B\in M_n(R)$ satisfying the property (*) All elements of the two-side ideal, in $M_n(R)$, generated by $AB-BA$, are nilpotent. McCoy showed that, if ...
4
votes
2answers
202 views

is 1/max(i,j) a bounded matrix on hilbert spaces?

I would like to know if the infinite matrix $[\frac{1}{\max(i,j)}]_{i,j\geq 1}$ represents a bounded operator on $\ell^2(\mathbb{N}^\star)$. It would be sufficient to know if the Lehmer matrix ...
1
vote
1answer
45 views

Maximising a Rayleigh quotient over a subspace II

Let $M$ be an SPD matrix and let $\Pi=QQ^T$ be the orthogonal projection onto the range of $Q$ (a "tall" matrix with orthonormal columns). I have an expression in the form $$\tag{1} ...
1
vote
1answer
66 views

Maximising a Rayleigh quotient over a subspace

Let $M\in\mathbb{R}^{n\times n}$ be symmetric positive definite and consider a matrix $Q\in\mathbb{R}^{n\times m}$ ($m<n$) with orthonormal columns ($Q^TQ=I$). I'm interested in finding an exact ...
4
votes
1answer
164 views

Vector Spaces of Symmetric Matrices of Low Rank

Let $K$ be a field, if necessary algebraic closed or of characteristic zero. Let $k$ be a positive integer. I am interested in linear subspaces $M \subseteq \textrm{Sym}_n(K)$, where ...
5
votes
1answer
245 views

Is the p-norm of a matrix strictly log-convex?

Let $A$ be a $n\times n$-matrix. We let $\|A\|_p$ denote the norm of $A$ when considered as a linear operator on $\ell^p(\{1,2,\ldots,n\})$, that is, $$ \|A\|_p = \sup_{x\neq ...
0
votes
1answer
87 views

Bounded domain matrix factorization

Assume we're trying to find $A\in [-1,1]^{n\times d}, B\in [-1,1]^{m\times d}$, from an observed matrix $C\in [-d,d]^{n\times m}$, where $C=AB^T$. The goal is to return $\widehat A, \widehat B$ such ...
0
votes
0answers
41 views

Spanning Hadamard product powers (Schur products) in Euclidean space

This is a question I asked last year over at stackexchange. Got no answer, it might be sufficiently non-trivial to justify bringing it into here. Fix two $k$-vectors $\mathbf u$ and $\mathbf v$, and ...
1
vote
0answers
68 views

Inverse of matrix of generalised harmonic numbers

For $s=0,1,\dots$ and $n=1,2,\dots$, denote $r_{n,s}=\sum_{k=1}^n k^s$. It is well-known that $r_{n,s}$ are polynomials in $n$ with leading term $\frac{1}{s+1}n^{s+1}$. Let $R_{n,s}$ be the ...
10
votes
1answer
257 views

doubly-stochastic isomorphisms of graphs

A doubly stochastic matrix that commutes with the adjacency matrix of a graph is a doubly-stochastic automorphism of that graph (definition by Tinhofer 1986). Each (classical) automorphism of a graph ...
6
votes
2answers
254 views

Does small Perron-Frobenius eigenvalue imply small entries for integral matrices?

Suppose that $M$ is an $n \times n$ matrix where each entry is a positive integer. Then $M$ is Perron-Frobenius and so has unique largest real eigenvalue $\lambda_{\textrm{PF}}$. Does an upper ...
1
vote
1answer
77 views

Bound on the 2-norm of a “special” matrix

Let $S\in\mathbb{R}^{n\times n}$ be such that $\|S\|_2\leq 1$, $P\in\mathbb{R}^{n\times m}$, $m<n$, with orthogonal columns ($P^TP=I$) so that $PP^T$ and $I-PP^T$ are orthogonal projectors, and ...
1
vote
2answers
254 views

when will the surfficient large power of a rational matrix be a integer matrix?

$A$ is a $n\times n$ matrix whose elements are all non-negative rational numbers and $Det(A)$ is a non-zero integer.Under what condition the following is true?(0) There exist a positive integer $M$ ...
2
votes
1answer
83 views

General method for under and over determined systems?

Suppose I have a system: $$ Ax = b $$ where $A$ is a $m$ by $n$ matrix which is less than full rank (neither full column nor row rank). In my particular case $m<n$. I'd like a combination of a ...
0
votes
1answer
96 views

Parametrization of real diagonalizable matrices with given eigenvalues

Complex diagonalizable matrices with given eigenvalues can be conveniently parametrized as $A=T^{-1} \Lambda T$, where $T$ is any invertible matrix, and $\Lambda=diag(\lambda_1,...,\lambda_N)$ with ...
12
votes
2answers
208 views

Order-preserving operator norms

Let us regard the $n\times n$ matrices as operators on the $n$-dimensional $\ell_p$ space; that is, we consider them as linear operators $\ell_p^n\to \ell_p^n$. When $p=2$, $M_n$ is a C*-algebra and ...
3
votes
1answer
144 views

Derivative of trace of pseudo inverse

Given three matrices $A$ (broad), $B$ and $C$, I'd like to find the derivative of \begin{align} f = \textrm{tr} \{BA^+\} + \textrm{tr} \{B(A^+)^TCA^+B^T\} \end{align} with respect to $A$, where ...
-2
votes
1answer
90 views

Is dual cone unique? [closed]

Suppose we have the following relationship, note that $A,B,C$ are closed convex matrix cones, $A^\ast=C,$ $B^\ast=C,$ can we state that $A=B$? Is the dual cone of a cone is unique? the definition ...
5
votes
1answer
248 views

Grassmann-Plücker relations for permanents

Let $K$ be a field, $1 \leq d \leq n$ integers and $V$ an $n$-dimensional vector space. The Grassmann-Plücker relations are quadratic forms on $\wedge^d V$ whose zero set is exactly the set of ...
1
vote
0answers
64 views

Dominant eigenvalue of sum of tridiagonal and diagonal matrices

Suppose I have a tridiagonal square matrix $A$ of some nice form, for which I know the eigenvalues $\lambda_1<\dots<\lambda_n$. $A$ is also essentially nonnegative (nonnegative everywhere except ...
5
votes
1answer
242 views

What it is the volume of the unit ball section of the cone of positive definite matrices?

Let $PD_{n}$ be the cone of positive definite $n \times n$ real matrices and let $B$ be the unit sphere in $n \times n$ dimensions. What is the volume of $PD_{n} \cap B$? EDIT: Let's assume that $B$ ...
0
votes
1answer
69 views

Eigendecomposition of a summation of matrices [duplicate]

Can anyone tell me if there's a way to relate the eigendecomposition of the result of a summation of matrices with the eigendecomposition of those matrices? More specifically: If I have a matrix $K = ...
6
votes
1answer
269 views

Diagonalization for sums of Hermitian matrices

I found an interesting question about diagonalizable matrices, Let $A,B\in \mathcal{M}_n(\mathbb{C})$ Hermitian, such that $AB\neq BA$. Do there exist complex numbers $u\neq v$, such that $A+uB$ and ...
2
votes
1answer
116 views

Symplectic block-diagonalization of a complex symmetric matrix

This is a follow-up question to the one asked here: Given a complex symmetric $2n\times2n$-matrix $A$, i.e., $A\in \mathbb{C}^{2n\times2n}$ with $A = A^T$. Is it possible, to block-diagonalize $A$ ...
4
votes
2answers
234 views

Lie's Theorem in characteristic $p$

Let $K$ be an algebraically closed field with characteristic $0$ and $V$ be a Lie sub-algebra of $M_n(K)$, the $n\times n$ matrices over $K$. If $V$ is solvable, then, according to Lie's theorem, $V$ ...
3
votes
2answers
191 views

Stabilization of the pencil of skew symmetric matrices by the orthogonal group

Good morning everybody. During my researches I've come across the following question. Let $A$ and $B$ be a couple of square $k\times k$ skew symmetric matrices on $\mathbb R$. Let us consider the ...
23
votes
1answer
613 views

Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$

Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle. Assume that the eigenvalues ​​of $A$ are included in a circle arc of ...
0
votes
0answers
37 views

Multiplicity of Minimum Eigenvalue of a Convex Combination of Hermitian matrices?

Let $A_1,\dots,A_L$ be $N\times N$ hermitian matrices. Consider the problem \begin{align} \lambda^{\star}=\max_{}&\lambda_{min}\left(\sum_{i=1}^{L}r_iA_i\right) \\ &r_i\geq ...
4
votes
1answer
123 views

Characterization(?) of coersive(?) elements in the special linear group

Take your favorite matrix norm $\|\bullet\|$ (my favorite is the Frobenius norm $\|A\| = \sqrt{\operatorname{tr} A A^t}$). Now consider the set $S_x$ of matrices $A,$ such that $\|A\| < x$ and ...
1
vote
0answers
85 views

Trace of Inverse matrix from Cholesky

This question was somewhat answered here: Fast trace of inverse of a square matrix. However, I feel like there was no complete answer wrt the Cholesky case. I have the matrix $\Sigma=LL^T$. Is there ...
3
votes
1answer
137 views

About partial uniqueness of SVD

In order to prove non-uniqueness of singular vectors when a repeated singular value is present, the book (Trefethen-Bau, considered the most authotitative book on the subject), argues as follows: Let ...
5
votes
0answers
246 views

Symmetric matrices with $\rho(A)\gg\|A\|_\infty$

For a symmetric real matrix $A$, denote by $\rho(A)$ the spectral radius of $A$, and by $\sigma(A)$ the largest absolute row sum of $A$; that is, $\sigma(A)=\max_i \sum_j |a_{ij}|$, where $a_{ij}$ are ...
2
votes
0answers
150 views

Norm bound of a complex resolvent

A well known result by Varah states that if $A$ is a strictly diagonally dominant matrix of dimension $n$, then $\|A^{-1}\|_{\infty} \le \max_i\frac{1}{|a_{kk}|-\sum_{j \neq k}|a_{kj}|}$, where the ...
3
votes
1answer
92 views

submatrix of a given size with maximum frobenius norm

Let $I\subset \{1,2,\ldots,n\}$, and let $|I|$ denote its cardinality. Now given a Hermitian matrix $\mathbf{A}\in\mathbf{C}^{n\times n}$. I am interested in finding the subset $I$ that maximizes the ...
5
votes
1answer
117 views

An inequality involving the spectral norm of a complex matrix

Let $A,B \in {M_n}(R)$ be real $n \times n$ matrices and let matrices $|A|$ and $|B|$ contain the absolute values of the elements of $A$ and $B$ respectively. Construct the complex matrices $C = A + j ...