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.
3,105
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Possible values of eigenvalues of Hadamard product of Hermitian matrices
One of the most important (and very well-known) result in the study of the spectrum of Hermitian matrices is Horn's conjecture (or theorem?), which provides a complete answer to the following problem:
...
0
votes
1
answer
166
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What is the following (matrix) operator called?
Let $\mathbf{A}=\begin{bmatrix} A_{11} & A_{12} & \cdots & A_{1m} \\ A_{21} & A_{22} & \cdots & A_{2m} \\ \vdots & \vdots & \cdots & \vdots \\ A_{m1} & A_{m2} &...
5
votes
0
answers
354
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Non-linear positive map
In the paper titled "Nonlinear completely positive maps" M. D. Choi and T. Ando extended natural definition of completely positive maps ignoring the linearity condition (Aspects of positivity in ...
1
vote
0
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204
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Complexity of reordering a matrix which consists independent sub matrices
Introduction:
Given a matrix A of a $k$ regular graph G. The matrix A can be divided into 4 sub matrices based on adjacency of vertex $x \in G$.
$A_x$ is the symmetric matrix of the graph $(G-x)$, ...
0
votes
1
answer
400
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SO(3) transformation that produces a reflection [closed]
This came up doing some research in quantum information. Let us consider two orthogonal three-dimensional unit vectors $v$ and $w$
$v^T\cdot w=0$,
and the Householder transformation
$H=I_{3}-2v\...
2
votes
1
answer
146
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Finding matrices $A$ such that the entries of $A^n$ have specified signs
What techniques are there for ensuring nonnegativity of various entries of matrix powers?
Specific Question: Consider a matrix $A\in SL_2(\mathbb R)$. Let $(A^n)_{i,j}$ denote the $(i,j)$ entry of ...
3
votes
1
answer
190
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Actions of the unit circle on finite complex matrices
Let $M_2(\mathbb{C})$ be the algebra of $2\times 2$ complex matrices and $\mathbb{S}^1$ the unit circle.
How many actions of $\mathbb{S}^1$ on $M_2(\mathbb{C})$ exist (up to isomorphism)? And on $...
1
vote
2
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465
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Worst case difference in rank by column-row swapping
Given a matrix $m\in\{-1,+1\}^{n\times n}$. Consider $m^\sigma$ to be collection of all matrices obtained from $m$ by permuting rows and columns.
Consider $\mathscr{M}[m^\sigma]$ to be collection of ...
3
votes
0
answers
113
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Rank of a particular matrix
Denote $P_{2n}$ to be collection of homogeneous total degree $2$ real polynomials in exactly $2n$ variables such that coefficient of every monomial is either $1$ or $-1$.
Split variable set into ...
6
votes
3
answers
414
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Probability a random matrix contains a short integer vector in its kernel
Consider a random $m$ by $n$ matrix $M$ with $m \leq n$, chosen uniformly over all those whose elements are in $\{0,1\}$ (or $\{-1,1\}$ if it is any easier). Is there any mathematical theory that ...
1
vote
1
answer
624
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Comparison of Lp norm of matrix and its entry wise norm. [closed]
I need to know the relation between operator norm of a matrix i.e. $ \Vert A\Vert_p$ for case of p=1 and 2 and its entry wise Frobenius norm $ \Vert A\Vert_F$.
10
votes
1
answer
521
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Expected size of determinant of $AA^T$ for random circulant and Toeplitz matrices
If $A$ is chosen uniformly at random over all possible $n$ by $n$ Toeplitz (or circulant) (0,1)-matrices, can we give any bounds for the expected size of the determinant of $AA^T$? All arithmetic is ...
1
vote
1
answer
943
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Expected size of determinant of $AA^T$ for non-square random $A$
If $A$ is chosen uniformly at random over all possible $m \times n$ (0,1)-matrices, what is the expected size of the absolute value of the determinant of $AA^T$. We can assume $m < n$ and all ...
1
vote
1
answer
135
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How to determine an unitary operator involved in an unitary transformation?
Let two real matrices $A$ and $B$ be unitarily equivalent. How to determine (computationally or theoretically) the unitary operator $U$ s.t. $A = UBU^\dagger$? Is it possible for some special class of ...
5
votes
2
answers
4k
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Determinant of block tridiagonal matrices
Is there a formula to compute the determinant of block tridiagonal matrices when the determinants of the involved matrices are known?
In particular, I am interested in the case
$$A = \begin{pmatrix} ...
3
votes
1
answer
442
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Largest symmetric matrix given rank
Let $\mathscr{M}[n,d]$ be collection of $n\times n$ symmetric matrices with real entries from $\{0,1\}$ such that every row/column is distinct with sum of every row/column being $d$.
What is minimum ...
35
votes
3
answers
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A curious determinantal inequality
In my study, I come across the following curious inequality, which I do not know a proof yet (so I am asking it here).
Let $A, B$ be $n\times n$ (Hermitian) positive definite matrices. It is very ...
2
votes
1
answer
81
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Information on special matrices similar to Jacobi matrices
Jacobi matrices are well known and deeply investigated mathematical objects from various point of view. One can arrive at these operators while studying discrete systems of particles interacting with ...
2
votes
2
answers
4k
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Find inverse and determinant of a symmetric matrix - for a maximum-likelihood estimation
Evaluate the determinant $\det \Omega $ and find the inverse matrix $\Omega^{-1}$ of:
$$\Omega = \begin{bmatrix} \beta_1^2(1+\theta_1^2) & \beta_1 \beta_2 & ... & \beta_1 \beta_{k-1} &...
3
votes
1
answer
253
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Prove or disprove a matrix logarithm equation
Let $\Bbb{S}_{++}^n$ denote the space of symmetric positive definite (SPD) $n\times n$ real matrices, and let $A,B\in\Bbb{S}_{++}^n$.
Is it possible to express the logarithm of $A^{-1}B$ as a ...
1
vote
0
answers
132
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Minimum rank of certain matrices
Let $\mathscr{M}[n]$ be collection of $n\times n$ matrices with real entries from $\{0,1\}$ such that every row is distinct and every column is distinct.
What is minimum real rank of matrices in $\...
5
votes
2
answers
434
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Spectral radius of a rank-1 perturbation
Suppose that $\bf A$ is an $n \times n$ matrix, and $\bf u$ and $\bf v$ are vectors. The matrix determinant lemma lets us easily compute the determinant of ${\bf A} + {\bf u} {\bf v}^\top$, while the ...
0
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2
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1k
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Representation Theory of $U(N)$
(1) Is it true that the category of representations of $U(n)$ is equivalent to the category of representations of $SU(N) \times U(1)$? If so, how is it proved, or what is a good reference. (I guess ...
3
votes
1
answer
527
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Determinant of a Certain Positive-Definite Block Matrix
Is there a lower bound for the determinant or minimum eigenvalue of the following $d$ by $d$ matrix in terms of $d$?
$$\Gamma=\left( {\begin{array}{cc}
I & B \\
B^{*} & I \\
\end{array} ...
1
vote
0
answers
1k
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comparing norms of block-matrices
Let $k\in\mathbb{N}$ and let $d\in\mathcal{M}_k=\mathcal{B}(\ell_2^k)$ be a $k\times k$ diagonal matrix with strictly positive entries. Let now $x_1,\ldots,x_m\in\mathcal{M}_k\,\,(m\in\mathbb{N}$ - ...
6
votes
1
answer
189
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Monte-Carlo computation of the Smith normal form
Quite some time ago I saw an article where a Monte-Carlo algorithm for computing the Smith normal form of an integer matrix was described. In this article the following problem was posed:
Suppose $P, ...
2
votes
1
answer
198
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Selecting columns of a set of boolean matrices with constraint on the ones in each row
I've come up with the following question in my research: Let $S$ be a finite set of $n \times n$ matrices with elements 0 or 1. denote $n_i$ as the total number of 1's in the $i$th row of all matrices ...
3
votes
3
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320
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Finding the square root of a special matrix
Let $n\in \mathbb N$ be a natural number, $x_1,\cdots.x_n$ be formal variables. Consider the following $n\times n$-matrix $M_n:=diag\{x_1^2+\cdots+x_n^2,\cdots,x_1^2+\cdots+x_n^2\}$, can we find a ...
2
votes
1
answer
519
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Vanishing of permanent of a Vandermonde matrix [Edited]
Does there exist an explicit criterion (or a good sufficient condition) for proving that a Vandemonde matrix:
$$A(x_1,\dots,x_n):=\left[ \begin{array}{llll}1 & x_1 &\dots& x_1^{n-1}\\ 1 &...
3
votes
1
answer
309
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Is there a systematic relation between the generating functions for the rows vs that for columns of infinite sized Carleman-matrices?
(I've asked this in MSE but nobody had an idea since dec 14...)
(Roughly related, but generalizing, of this earlier MSE question)
Background: The ...
1
vote
1
answer
217
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Row-stochasticity of the Jacobian matrix of a stationary distribution
Let $P_{\mathbf{p}}$ be a $n \times n$ row-stochastic matrix whose entries are a function of a probability vector $\mathbf{p} \in \mathbf{R}_{> 0}^n$, $\sum_i p_i = 1$ and define the following ...
1
vote
1
answer
531
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Partial Vandermonde circulant determinant expression
Consider following partial Vandermonde type, circulant matrix
$\begin{bmatrix}
x_1 & x_2 & 0 & \dots & 0 & x_n\\
x_1^2 & x_2^2 & x_3^2 & \dots & 0 & 0\\
\vdots ...
1
vote
0
answers
146
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Perturbation of eigenvalues of some special matrices
In perturbation theory of linear operators, one major question is how the eigenvalues of a linear operator $A$ change under a small perturbation, $A(x) = A + xP$, with $x\in\mathbb{R}$. For instance, ...
4
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0
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Relation between Euclidean distance matrices and squared-distance matrices
Let us refer to $D$ as the matrix with $D_{ij} = ||x_i - x_j||_2$ and $H = D \circ D$ e.g. the matrix with $H_{ij} = ||x_i - x_j||_2^2$ as its entries. Are the spectra of these matrices related at all?...
8
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1
answer
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Computation time of Smith normal form in Maple
I am using Maple to compute the Smith normal form (SNF) of a $120 \times 120$ matrix and it seems that I will never get an answer back. I have checked my code for small cases and I believe that it is ...
0
votes
0
answers
197
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Finding a "special" non singular submatrix
Given a square integer matrix $A \in M_n(Z)$ and two subsets $I, J \subset \{ 1, \ldots, n\}$, we define $A_{I,J}$ as the sub-matrix of $A$ containing the rows (resp. columns) whose index is in $I$ (...
6
votes
1
answer
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Eigenvectors as continuous functions of matrix - diagonal perturbations
The general question has been treated here, and the response was negative. My question is about more particular perturbations. The counterexamples given in the previous question have variations not ...
2
votes
0
answers
36
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Is there some kind of lower bound for estimation error of the estimation of (near) low-rank matrices in high-dimension?
I'm reading S.Negahban and M.J.Wainright's paper, ESTIMATION OF (NEAR) LOW-RANK MATRICES WITH NOISE AND HIGH-DIMENSIONAL SCALING. In the paper, they give a upper bound for estimation error of the ...
7
votes
2
answers
251
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What methods do we have to understand the spectrum of matrices with restricted entries?
Consider questions of the form (or the "most probable value of" version of these questions rather than the "largest possible"),
What is the largest possible spectral radius of a $...
7
votes
1
answer
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Determinant of some covariance matrix (Gaussian kernel process)
Let $x_1,\dots,x_p$ be $p$ points in $\mathbb{R}^n$ ($n\geq 2$) with $x_1=0$. Consider the symmetric matrix $M(x)=(m_{ij}(x))_{1\leq i,j\leq p}$ where $m_{ij}(x) = \exp(-\frac{1}{2}\Vert x_i - x_j\...
0
votes
1
answer
125
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Reference for measures of commutativity needed
I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if A and B are two n×n Hermitian matrices, and [A,B]=C.
I'd like a function μ:Cn×n→[0,∞) ...
9
votes
1
answer
912
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Convexity of the product of two exponential matrices
Let $S\subset\mathbb{R}$ be a convex set and $\mathbb{S}^{n}$ be the set of real symmetric matrices of order $n\times n$.
A matrix valued function $\Gamma: S \rightarrow \mathbb{S}^{n}$ is said to ...
1
vote
0
answers
84
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Smallest sum of original column entries in 2d matrix [closed]
I have an interesting optimization problem I am trying to solve now and I thought I'd share it here in order to find the best answer. The problem itself is not complicated and it is stated like this:
...
1
vote
1
answer
82
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Given $M$, minimize $|Mx|_0$
Given a matrix $M \in \mathbb{R}^{n \times m}$, I would like to find
$\min_{x \in \mathbb{R}^m} \|Mx\|_0$ such that $x \neq 0^m$,
where the $\ell_0$ "norm" is measured by simply counting the number ...
0
votes
0
answers
86
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Derivative of a conjugation of matrices
Let $\mathcal{M}_n$ be the space of complex $n\times n$ matrices. Let $\Phi\colon \mathbb{D}\to \mathcal{M}_n$ and $\psi \colon \mathbb{D}\to \mathcal{M}_n$ be holomorphic functions. Consider the ...
1
vote
0
answers
249
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Is there a way to simplify this apparently huge characteristic polynomial calculation?
Say I am given the $0/1$ adjacency matrix of an undirected graph. Also I am given a representation $\rho$ of some group $G$ and an orientation has been arbitrarily chosen along each edge. Let $E^{...
12
votes
5
answers
2k
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Analogue of Cayley Hamilton theorem for operators on Hilbert space
Is there an analogue of Cayley Hamilton theorem which holds for operators on a separable Hilbert space. Obviously the characteristic polynomial will be replaced by something else.
9
votes
2
answers
1k
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On closest unitary matrix
In this question $\|A\|_p$ is the normalized $p$-th Schatten norm which is defined to be $\left(\mathbb E_{i} \lambda_i^p\right)^{1/p}$, where $\lambda_i$ are singular values of matrix $A$.
Suppose ...
63
votes
7
answers
9k
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How to prove this determinant is positive?
Given matrices
$$A_i= \biggl(\begin{matrix}
0 & B_i \\
B_i^T & 0
\end{matrix} \biggr)$$
where $B_i$ are real matrices and $i=1,2,\ldots,N$, how to prove the following?
$$\det \big( I + e^...
2
votes
0
answers
169
views
Optimization with random matrix
Consider $J$ a random matrix of size $n\times n$ with i.i.d. Gaussian entries $J_{ij} \sim \mathcal{N}(0,\sigma^2/n)$. Let $f(x)=tanh(x)$, and for $x\in\mathbb{R}^n$, $f(x)$ denotes the vector where $...