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
20 views

system of linear equations [on hold]

These are the two known equations (I2+I3)-(I1+I4)/(I1+I2+I3+I4) = 2x/L (I2+I4)-(I1+I3)/(I1+I2+I3+I4) = 2y/L where I know x,y,L values. How can I find the values of I1,I2,I3,I4?
0
votes
0answers
39 views

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 $I$ ...
3
votes
1answer
72 views

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 ...
-4
votes
0answers
31 views

Large Matrix problem [on hold]

I have been searching the internet for a large matrix problem but didn't find any problem. I am searching for a problem like for example Kirchhoff's Rules Problems where you have I1, I2...etc ...
2
votes
0answers
13 views

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 ...
-2
votes
0answers
24 views

Matrix decomposition ( Kronecker product decomposition) [closed]

How to solve the following matrix equation: $Q = A \bigotimes B$ (Given $Q$)? Can we prove that if $Q \in \mathbb{R}^{n^2\times n^2}$ and $Q$ is symmetric, there always exist $A,B$ such that $Q = A ...
6
votes
2answers
189 views

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 $n \times n$ matrix ...
-2
votes
0answers
55 views

Rotation Matrices [closed]

I am not a mathematician strictly speaking I am in fact a computer science PhD student but my problem couldn't be solved by computer scientist I'm sure. I am actually programming in Android. An ...
7
votes
1answer
96 views

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 - ...
0
votes
1answer
68 views

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,∞) ...
0
votes
0answers
21 views

Orthogonal matrices [migrated]

I've always been told that special orthogonal group (matrices M such as M$M^T$=I and detM=1) represents the rotation. But how do you prove it? I thought it was easy until i tried to do it for ...
6
votes
1answer
189 views

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
0answers
52 views

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
1answer
55 views

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
0answers
55 views

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
votes
0answers
19 views

Matrix Calculus and Dot Products [migrated]

I'm a beginner at matrix calculus and I've been using the Wikipedia page for my rules. I'm having some trouble with this problem that I've faced. I know the solution, but I'm wondering why my strict ...
-1
votes
0answers
29 views

Generating pairs of $\mathfrak{su}(N)$ and $\mathfrak{so}(N)$ Lie algebras - characterization

It is known that any simple Lie algebra can be generated by two elements (Kuranishi paper). Is it know how to characterize ALL generating pairs for $\mathfrak{su}(N)$ and $\mathfrak{so}(N)$ algebras?
0
votes
0answers
61 views

prove that a function is approximatively three dimensional

Let $D_n(x)$ be a diagonal matrix of size $N\times N$ where the $k$th element is $\exp(2\pi\jmath x(n+(k-1)/N)$. Let $P_n$ be a random diagonal $N\times N$ matrix where each diagonal element is a ...
1
vote
0answers
154 views

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 ...
6
votes
4answers
430 views

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.
2
votes
0answers
295 views

How to prove the following determinant identity? [migrated]

This problem is relevant to the spin operator matrix elements in the quantum 1D XY model. For any even integer $N$, define two sets ...
8
votes
2answers
328 views

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 ...
41
votes
7answers
2k views

How to prove this determinant is positive?

Given the 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 that $\det(I + ...
1
vote
0answers
44 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 ...
2
votes
2answers
246 views

Is anything known about the eigenspectrum of the regular representation of the permutation group?

I am looking for information like upper bounds on how many times any eigenvalue can occur or something like how many eigenvalues can be there in some given range. Is anything like this known? The ...
0
votes
0answers
43 views

Centralizer of a non-regular Lie algebra element

It is well understood that the centralizer of a regular element $A$ of a Lie algebra of complex (square, diagonalizable) matrices consists of polynomials $p(A)$ in that element of degree less than $n$ ...
1
vote
0answers
24 views

LU growth factor applied to LDL of a Positive Semidefinite matrix [closed]

For a Positive Semidefinite matrix $A$, which we can decompose through $LDL$ decomposition as follows: $A=LDL^\text{T}$; how can we prove that for a decomposition $A=LU=L(DL^\text{T})$, the growth ...
0
votes
0answers
22 views

Degree $2$ nilpotent matrices with non-zero product [migrated]

Let $n$ be sufficiently large positive integer. Let $M_i$ be $n$ matrices with entries in either $\mathbb{C}$ or $\mathbb{Z}/n \mathbb{Z}$. Is it possible, (1) for $1 \le i \le n, M_i^2=0$ and ...
1
vote
0answers
72 views

Is my particular finite dimension Toeplitz matrix always strictly positive?

Let $H(\omega), \; -\pi \leq \omega \leq \pi$ be a real-valued function with a continuous band of zeros, that is (for simplicity) $H(\omega)=0, \; |\omega|\geq \beta \pi$. Define a sequence of banded ...
2
votes
0answers
178 views

An (open?) problem about a sequence of nested principal sub-matrices and their determinants

Problem: Let $A$ be a $n \times n$ integer matrix, $\det(A) = \pm 1$. Under which conditions there exist a nested sequence of principal submatrices of size $n$ such that they all have determinant $\pm ...
4
votes
0answers
89 views

Geometric interpretation of the Desnanot-Jacobi Identity

Given a square $n\times n$ matrix $M$, let $M_i^j$ denote the $(n-1)\times(n-1)$ matrix obtained from M by omitting the i-th row and j-th column of $M$. The Desnanot-Jacobi Identity states ...
2
votes
1answer
223 views

Lebesgue measure of set of $y\in\mathbb{R}^n$ such that $x,y,Ay$ are linearly dependent

I've asked this question here on math.stackexchange, but I have been unable to solve this yet, so I'm hoping I can get some advice here. Consider a vector $x\in \mathbb{R}^n$ and a real $n\times n$ ...
2
votes
1answer
69 views

Does Schatten-p (quasi-)norm satisfy the norm inequality for 0<p<1?

I'm reading the paper by ANGELIKA ROHDE AND ALEXANDRE B. TSYBAKOV, ESTIMATION OF HIGH-DIMENSIONAL LOW-RANK MATRICES. And in the paper, they provide an inequation of the Schatten-p (quasi-)norm, ...
7
votes
0answers
79 views

What is the symmetry group fixing norms of elements of a unitary matrix?

Let $N\geq1$ be an integer and let us say that two matrices $U,V\in U(N)$ are related if $|U_{ij}|=|V_{ij}|$ for all indices $1\leq i,j\leq N$. When exactly are two unitary matrices related in this ...
0
votes
2answers
64 views

Symmetric matrix from a nonsymmetricc matrix

Basically this is a part of a long algorithm to calculate some matrix properties. Given an upper triangular square matrix R, how can I find an orthonormal matrix W (possibly iteratively) such that WR ...
1
vote
2answers
126 views

Norm of a matrix operator with a special structure

Let $\{\alpha_{n}\}_{n\in\mathbb{N}}$ be positive sequence such that $$\sum_{n=1}^{\infty}\alpha_n<\infty.$$ Question: Is there any chance to evaluate the operator norm of the matrix operator ...
1
vote
1answer
161 views

Homotopy type of certain maps on complex grassmanian

$G(k,n)$ is the complex grassmanian which is homeomorphic to the space of projections in $M_{n}(\mathbb{C})$ with trace $k$. So we can Identify $G(k,n)$ with $$\{A\in M_{n}(\mathbb{C})\mid ...
1
vote
1answer
70 views

On the least singular values

Let $A$ be a square matrix of size $n \times n$ ($n>2$) and let $B$ be $A$ if we delete the last row and column (size $(n-1) \times (n-1)$). Let $\sigma (A)$ be the least singular value of $A$ and ...
2
votes
4answers
286 views

Finding commuting matrices

Is there a procedure for finding all matrices which commute with two given square and complex matrices? For example, given two elements $A,B \in$ $\mathfrak{su}(4)$ is it possible to find all ...
2
votes
1answer
83 views

Equivalence of entrywise 1-norm and Schatten-1 norm

Let $A \in \mathbb{R}^{m\times n}$ and $\|A\| = \sum_{i, j} |A_{i,j}|$. I am looking for constants $\alpha, \beta \in \mathbb{R}$ such that $\alpha \|A\| \leq \|A\|_* \leq \beta \|A\|$ The function ...
1
vote
2answers
125 views

A certain matrix associated to graphs

I am not very familiar with graph theory, but I need some results for my work. Thus, the question is, whether the following has already been studied and where I can find it. Let $G=(V,E)$ be an graph ...
2
votes
1answer
58 views

Decomposing large symmetric banded sparse matrices

I'm investigating 3D image deblurring and one of the approaches I'm interested in is applying spatial regularisation. To do this I have generated a matrix $A$ which encodes the 6-connectivity of each ...
0
votes
1answer
105 views

Minimal dimension of a Lie algebra of matrices, with a restrictive property

Let $\mathfrak{g}$ be a sub-Lie-algebra of $\mathfrak{gl}_n(\mathbb{C})$, the Lie algebra of complex $n\times n$ square matrices. Let us call $(H)$ the hypothesis: for all $x, y\in\mathbb{C}^n$, ...
0
votes
0answers
28 views

Is there a term for “ranked distance” matrices?

In a n by n "ranked distance matrix" each element has a rank $r_{ij}$ between 1 and n that indicates it is the $r_{ij}$th smallest element in column $i$ of a corresponding Euclidean distance matrix. ...
1
vote
1answer
100 views

Dense symmetric unitary integer matrix?

Can someone give me a nontrivial example of a symmetric unitary integer matrix? I'm looking for something as dense as possible (i.e., not too many 0's); 5 <= size <= 8 would be ideal.
12
votes
1answer
205 views

How can we interpret the eigenvalues and eigenvectors of Euclidean Distance Matrices?

I asked this question in Math Stack Exchange earlier here: http://math.stackexchange.com/questions/1199380/what-is-the-intuition-behind-how-can-we-interpret-the-eigenvalues-and-eigenvec and since I ...
6
votes
5answers
318 views

The maximal eigenvalue of a symmetric Toeplitz matrix

Let $0\le x\le 1$ be a real number. Denote by $A_n(x)=(a_{ij})$ the $n$ by $n$ matrix such that $a_{ij}=x^{|i-j|}$ and let $\lambda_n(x)$ be the maximal eigenvalue of $A_n(x)$. Is there any ...
0
votes
0answers
85 views

What is wrong with the argument that zero permanent is polynomial?

This Lecture summarizes some well known facts about $\#P$ completeness of permanent. Given a CNF formula $\phi$ on $n$ variables, they construct matrix $A$ such that: $$perm(A)=4^{3m} \#SAT(\phi)$$ ...
4
votes
1answer
172 views

Represent matrix immanants using Schur functions

For each irreducible character $\chi^\lambda$ of the symmetric group $S_n$, the immanant of an $n\times n$ square matrix $A$ is defined as \begin{equation*} d_\lambda(A) := \sum_{\sigma \in S_n} ...
7
votes
0answers
269 views

Does this inequality always hold?

Denote the adjacency matrix of a given undirected graph by $g$. It is an $n$-by-$n$ symmetric Boolean matrix with elements on the diagonal to be zero ($n\geq 3$). Let $g_{12}=g_{21}=g_{13}=g_{31}=1$ ...