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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.

2
votes
1answer
118 views

Maximum rank in a class of $0\,$-$1$ partitioned matrices satisfying combinatorial constraints

We are given a matrix $M \in \{0,1\}^{n\times n}$ satisfying the following property. The rows and columns of $M$ can be partitioned into $k$ rowgroups and $k$ colgroups respectively, such that in ...
-4
votes
1answer
51 views

How to calculate $y^T \mbox{diag}(A^T B A) \,y$ efficiently? [closed]

I want to calculate $$y^T \mbox{diag}(A^T B A) \,y$$ where $y$ is a $n \times 1$ vector. $A$ is a $m \times n$ matrix where $n \gg m$. $B$ is a $m \times m$ symmetric positive definite matrix; the ...
7
votes
1answer
214 views

Lower bound on the eigenvalues of the Laplacian

I am looking for a graph for which $2 d_{i} < \mu_{i}$, for some index $i$, where $\mu_{1} \leq \mu_{2} \leq \dots\leq \mu_{n}$ are the eigenvalues of the Laplacian matrix $L(G)$ and $d_{1} \leq d_{...
3
votes
1answer
199 views

« Generalized simultaneous diagonalization » of a pair of symmetric, non-commuting, positive semi-definite matrices

I hope my question is trivial for some of you but for the time being I’m lost somewhere between the generalized eigenproblem, simultaneous diagonalization of quadratic forms, simultaneous SVD, ...
2
votes
0answers
60 views

Points on Sphere whose image, under symmetric positive definite matrix, is contained in cube

Let $\Sigma \in \mathbb{R}^{n \times n}$ be a symmetric, positive definite matrix and let $\mu_r$ denote surface measure on the sphere in $\mathbb{R}^n$ with radius $r$. Let $$ R = \{x \in \mathbb{R}^...
14
votes
5answers
1k views

Matrix trace & norm [closed]

For any nonnegative semidefinite matrix $A$ and any matrix $B$, we have $$\mbox{tr} (AB) \le \mbox{tr} (A) \, \|B\|$$ where $\mbox{tr}(\cdot)$ is the trace and $\|\cdot\|$ is the operator norm. How ...
2
votes
1answer
71 views

How the eigenvalues change when a Hermitian matrix is left multiplied and right multiplied by a diagonal matrix?

Suppose there is a Hermitian matrix $S$ with eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_K$. There is a diagonal matrix $D$ whose entries on the main diagonal are positive. What are the ...
0
votes
0answers
102 views

Upper bound on matrix perturbation such that all eigenvalues lie within the unit circle

Consider the matrix $$N=\left[\matrix{\mathbb{I}_n-\epsilon L & X\\ \epsilon Y & Z}\right]$$ where $\epsilon>0$ is a small positive parameter and $Z$ is a square $m\times m$ matrix with ...
0
votes
0answers
46 views

On fixed dimension integer programming complexity when LLL is perfect

Fixed dimension integer programming (both Lenstra's and Barvinok's) uses LLL which guarantees short vectors but not the shortest possible. Suppose for a given integer programming problem find $x\in\...
2
votes
2answers
180 views

Regarding minimal elementary generators for $GL(n, \mathbb{Z})$

I have a result concerning the minimal number of elementary generators (and by this I mean generators which are elementary matrices) for $GL(3, \mathbb{Z})$. I'm currently working on extending the ...
1
vote
0answers
59 views

specific sequence of matrices making a strange ratio of matrix norms diverging

For any $t>0$ define $d_t:=\operatorname{diag}j^t=\operatorname{diag}(1^t,2^t,\ldots)$. Now pick up such a $t>0$ and an arbitrary $\theta\in\big(0,\frac12\big)$. For every $k\in\mathbb{N}$ find ...
0
votes
1answer
833 views

What is special about 2 + $\sqrt{3}$?

Well, one thing is special about it, but it takes a while to explain. Please let me know, whether this number occurs in other special occasions as well. The explanation: Let $p$ be a complex ...
2
votes
0answers
88 views

Representable integer matrices

Let $C, R \in \mathbb{Z}^n$. If there is an $n \times n$-matrix $M$ with all entries being integers such that the sum of the entries of column $k$ equals $C(k)$, and the sum of the entries of row $k$ ...
5
votes
1answer
130 views

Rational map given by pfaffians

Consider a general skew-symmetric $(n+1)\times (n+1)$ matrix $Z$, and let su map $Z$ to the point of $\mathbb{P}^n$ determined by $[pf_0(Z):\dots:pf_n(Z)]$ where the $pf_i(Z)$ are the principal ...
11
votes
2answers
264 views

Sum-regular $\{0,1\}$-matrices

Let $n\in\mathbb{N}$ be a positive integer. We say that an $n\times n$-matrix $A$ with all entries in $\{0,1\}$ is $k$-regular for some $k\in \{0,\ldots,n\}$ if the sum of every row and the sum of ...
0
votes
0answers
72 views

Complex symmetric matrices in a conjugacy class of SU(N)

Let $a$ be the following matrix in SU(N) $$ A=\begin{bmatrix}a & \\ & a\\ &&\ddots \\&&&b\\&&&& b\\&&&&&\ddots\end{bmatrix} $$ with $m$...
5
votes
3answers
148 views

How many $40$-vertex cubic bipartite graphs have determinant $\pm 3$?

To get some feel for the size of a particular computation, I would like to know the approximate number of (pairwise-nonisomorphic) cubic bipartite graphs on $40$ vertices whose bipartite adjacency ...
1
vote
2answers
81 views

Low-rank solution of generalized Sylvester equation

Let $n \in \mathbb{N}$. Let $A,B,C,D$ be non-singular $n \times n$ matrices. If the matrix pencils $A-\lambda C$ and $B-\lambda D$ are regular and have disjoint spectra, then $$AXB-CXD = 0$$ has a ...
0
votes
0answers
115 views

Convexity of the Frobenius norm of the product of matrices

I have a question similar to Convexity of the Frobenius norm of the product of two matrices. I am not able to comment on that question as I don't have enough reputation, and that is why I have asked a ...
6
votes
0answers
303 views

A rank inequality

Suppose $$M := \begin{bmatrix} M_{11} & \cdots &M_{1d} \\ \vdots & \ddots & \vdots \\ M_{d1} & \cdots & M_{dd} \end{bmatrix}$$ is a $d \times d$ block matrix such that $$M_{...
2
votes
0answers
70 views

How to prove this matrix is idempotent and that it obeys a telescoping identity

Let $P_k$ be the size $k$ leading principal submatrix of the real-valued $N \times N$, invertible matrix $M$, and let $Q_k$ be the size $N$ matrix having $P_k^{-1}$ in the top left corner, and zeroes ...
0
votes
0answers
44 views

Norm inequalities for operator valued matrices

I am looking for literature about operatorvalued matrices, especially norms and norm inequalites of these matrices (e.g. the entries are elements of the space of linear bounded operators $\mathcal{L}(...
8
votes
5answers
296 views

Nearest matrix orthogonally similar to a given matrix

Given $A,B\in\Bbb R^{n\times n}$ is there technique find $$\min_{ T\in O(n,\Bbb R)}\|A-TBT^{-1}\|_F\mbox{ or }\min_{ T\in O(n,\Bbb R)}\|A-TBT^{-1}\|_2$$ within additive approximation error in $\...
4
votes
1answer
157 views

A Handbook of Matrix Factorizations

I am looking for a good collection of facts regarding the various types of matrix factorizations, something like a "Handbook of Matrix Factorizations" or a very-thorough review paper. I am hoping for ...
21
votes
6answers
2k views

What are the possible eigenvalues of these matrices?

Edit: since we seem a bit deadlocked at this point, let me weaken the question. It's fairly easy to see that the set of 8-tuples of reals which can be the eigenvalues of a matrix of the desired form ...
0
votes
0answers
88 views

Number of full-rank matrices over a finite field with a full weight

I want to calculate the number of mxn matrices over a finite field GF(q) with m < n for which two conditions hold: The matrix has a full rank (m) There are no zero entries in the matrix, i.e every ...
3
votes
1answer
695 views

Eigenvalues and eigenvectors of tridiagonal matrices

What can I say about the eigenvalues and eigenvectors of the tridiagonal matrix $T$ given as $T = \begin{pmatrix} a_1 & b_1 \\ c_1 & a_2 & b_2 \\ & c_2 & \ddots & \ddots \\ &...
2
votes
0answers
54 views

Non singularity of a generalised Vandermonde matrix through Hadamard product

I'm currently trying to prove the following. Consider $k_1,\dots,k_N$ complex numbers not lying on the real and imaginary axes. Then consider \begin{equation} W_N(x)= \text{Wronskian}\big(\cosh(k_1x),...
3
votes
2answers
706 views

Proving that a matrix is positive semidefinite

Let matrices $A, B$ be positive semidefinite. Can we prove that $A(I+BA)^{-1}$ is positive semidefinite?
8
votes
2answers
256 views

Matrix exponential, containing a thermal state

This question was originally posted on MSE, and I'm cross posting it here. Define an infinite matrix $$ M = \begin{bmatrix} 0 & -1 & 0 & 0 & \cdots \\ 1 & 0 & -2 & 0 &...
9
votes
1answer
242 views

Regular $p$-norm of a matrix

Let $n \in \mathbb{N}$ and $p \in [1,\infty]$ be fixed and endow $\mathbb{C}^n$ with the $p$-norm $\|\cdot\|_p$. For every matrix $A \in \mathbb{C}^{n \times n}$ we denote the operator norm of $A$ as ...
0
votes
0answers
243 views

Eigenvalues of a specific Hankel matrix

I have an $\frac{N}{2} \times \frac{N}{2}$ matrix $G$ with entries given by \begin{equation} G_{ij} = \frac{1}{\sin(\frac{\pi}{N}(i+j-\frac{3}{2}))}, \;\;\;\;\;\;\;\; 1 \le i,j \le \frac{N}{2}, \end{...
0
votes
1answer
68 views

$A_{n \times m} D_{m \times m} A^T_{m \times n} + \alpha I_{n \times n}$

Assume that we have a matrix product of form $B=A_{n \times m} D_{m \times m} A^T_{m \times n} + \alpha I_{n \times n}$. $D$ is a positive diagonal matrix, $I$ is identity matrix, $\alpha>0$ and $m ...
5
votes
0answers
135 views

Resultant of a binomial and a trinomial

Does anyone know of any papers dealing with the resultant of a binomial $x^n+a$ and a trinomial $x^r+bx^s+c$ ? Even special cases would be of interest. (The resultant of two binomials is well known.)...
2
votes
1answer
134 views

Properties of Zero Line-Sum Matrices

By a Zero Line-Sum (ZLS) matrix I mean matrices with the property, that each row sum and each column sum equals zero: $$A\in\mathbb{R}^{m\times n}:\ \sum_{i=1}^{n}a_{ij}=\sum_{j=1}^{m}a_{ij}=0$$ ...
5
votes
1answer
224 views

On faces of convex sets of positive semidefinite matrices

A face $F$ of a convex set $C$ is a set such that, if $x \in F$ is a convex combination of other elements in $C$, then they must also be in $F$. I will denote by $F(C,x)$ the smallest face of $C$ ...
3
votes
1answer
129 views

A matrix monotonicity question

Let $X\in\mathbb{R}^{n\times n}$ be a positive semi-definite matrix and $A\in\mathbb{R}^{n\times n}$ be a stable matrix, i.e. a matrix whose eigenvalues are strictly inside the left-half complex plane....
2
votes
1answer
117 views

Quadratic matrix equation for $X\in \mathbb{C}^{ n\times p}$ with Hermitian parameters

Let $A\in \mathbb{C}^{ n\times n}$ and $B \in \mathbb{C}^{p \times p}$ be Hermitian matrices with $p < n$. Find matrix $X$ such that $X^*AX=B.$ Solution in the case of positive definite $A$ and $...
7
votes
3answers
252 views

Checking positive semi-definiteness of integer matrix

Key Problem : Is there any theorem about eigenvalues or positive semi-definiteness of small size matrices with small integer elements? I have to check positive semi-definiteness of many symmetric ...
0
votes
0answers
90 views

Gradient of the trace of the logarithm of a product

Suppose $G$ and $A$ are full rank matrices. Is there a closed-form solution for $$\nabla_G \mbox{Tr} (A \log GG^\top)$$ when $A$ is a PSD matrix?
3
votes
2answers
293 views

If $S$ is a nonsingular symmetric matrix over a number field and $D_k$ is its principal minor of order $k$, is $\frac{D_k}{D_{k-1}} > 0$ always true?

In Chapter II, Paragraph 4, Section 1 of F. R. Gantmacher, The Theory of Matrices. Vol. 1 English translation by K. A. Hirsch. Chelsea Publishing Company. 1959. SBN 8284-0131-4, the following ...
3
votes
1answer
331 views

Bounds for eigenvalues of block matrix

Let's say I have a block matrix of the form $$X = \begin{bmatrix} A & B\\ B^T & C\end{bmatrix}$$ where $A$, $C$, and $X$ are all positive definite. I have bounds on both the minimum and ...
11
votes
1answer
632 views

Determinant of identity matrix plus Hilbert matrix

I am looking for the determinant $$ \det(I_n + H_n) $$ where $I_n$ is the $n \times n$ identity matrix and $H_n$ is the $n \times n$ Hilbert matrix, whose entries are given by $$ [H_n]_{ij} = \frac{...
24
votes
2answers
866 views

Is this proof of Perron's theorem correct, and if so is it original?

A few years ago, I came up with this proof of Perron's theorem for a class presentation: http://www.math.cornell.edu/~web6720/Perron-Frobenius_Hannah%20Cairns.pdf I've written an outline of it below ...
0
votes
1answer
87 views

Does stability imply that the logarithmic norm is negative?

Let $A \in \mathbb R^{n\times n}$ and assume that all eigenvalues lie in the left open halfplane. Is it true that the logarithmic norm $\mu_2 (A):= \lambda_{\max} \left(\frac{A + A^T}{2}\right)<0?...
8
votes
1answer
298 views

Gaussian integrals over the space of symmetric matrices

Let $S\in\mathcal S_N$ be a $N\times N$ symmetric matrix over the reals, and introduce the (normalised) gaussian measure $$ \mathrm d\mu(S):=2^{-\frac 12N}\pi^{-\frac14N(N+1)}\exp\left[-\frac12\...
2
votes
2answers
540 views

Inverse of particular lower triangular matrix

I have an $n \times n$ lower triangular matrix $A$ where $$A_{i,j} = {\bf x}_i{\bf x}_j^H,\quad i>j$$ $$A_{ii}=1, \quad 1 \leq i \leq n,$$ and ${\bf x}_i$ is a $1 \times k$ (row) vector, where $...
3
votes
0answers
59 views

Maximum number of negative entries in a matrix with positive diagonal and given rank

Suppose $A \in \mathbb{R}^{n \times n}$ has positive entries on it main diagonal and $\mbox{rank}(A) =: d < n$. Then, what is the maximum number of many negative entries $A$ can contain? If, in ...
4
votes
0answers
78 views

A Toeplitz variant of the Hilbert matrix

It is well-known that the Hilbert matrix $H$, i.e., the symmetric Hankel matrix with entries $$H_{m,n}=\frac{1}{m+n-1}, \quad m,n\in\mathbb{N},$$ determines a bounded operator on $\ell^{2}(\mathbb{N}...
3
votes
1answer
174 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} ...