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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1
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0answers
28 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 ...
6
votes
7answers
637 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 ...
4
votes
2answers
954 views

rank-one perturbation of a matrix corresponding to a specific spectrum

Let $A$ be a real symmetric matrix whose spectrum is $\lambda_1,\lambda_2,\ldots,\lambda_n$. Let $A'$ be the matrix obtained by adding a perturbation to $A$. The requirement is that only the second ...
2
votes
2answers
220 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
32 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 $n$ ...
6
votes
1answer
6k views

vector to diagonal matrix [closed]

For any column vector we can easily create a corresponding diagonal matrix, whose elements along the diagonal are the elements of the column vector. Is there a simple way to write this transformation ...
1
vote
0answers
16 views

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

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 ...
-3
votes
0answers
44 views

What is a constant-rank matrix definition? [on hold]

I have had no luck finding a definition of constant-rank matrix. Even less luck with an intuitive example. My interest is to understand when can I apply a formula for the derivative of the ...
1
vote
0answers
59 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 ...
0
votes
0answers
15 views

Determinant of a 5x5 matrix [migrated]

I have a little problem with a determinant. Let $A = (a_{ij}) \in \mathbb{R}^{(n, n)}, n \ge 4$ with $$a_{ij} = \begin{cases} x \quad \mbox{for } \,i = 2, \,\, j \ge 4,\\ d \quad \mbox{for } ...
-4
votes
0answers
29 views

Invariance of absolute determinant under alternating sign changes in columns [closed]

I (experimentally) notice that for an $MN \times MN$ matrix, where $M$ is even if $N$ is odd and vice-versa, if I multiply each column $c_i$ by the elements of either (i) $T_1 = [t_1^{(1)}, ...
2
votes
1answer
331 views

Diagonalization of 4th order tensors

I have been wondering about the following problem... Let $n$ be a positive integer and denote by $M_n^s$ the space of symmetric $n\times n$ real matrices. Now, we look at the space $\mathcal ...
0
votes
0answers
27 views

Binary Operator and Vector Combination Problem [closed]

Given a set of vectors, a set of binary operators, and a solution constraint. Is there a mathematical way to combine the vectors and operators to attain the solution constraint? The vectors are all ...
21
votes
4answers
23k views

Eigenvalues of Matrix Sums

Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when they are Hermitian and positive-definite? I am investigating ...
2
votes
0answers
166 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 ...
2
votes
1answer
205 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 $n\times n$ ...
4
votes
0answers
76 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
61 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
66 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 ...
1
vote
2answers
121 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 ...
-4
votes
0answers
59 views

Elementary book on Matrices [migrated]

can anyone recommend a book that focusses on matrix theory at an elementary level? I was never taught matrices in high school (25 years ago) and I'm teaching myself algebra using The Everything Guide ...
1
vote
1answer
160 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 ...
25
votes
1answer
748 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 ...
6
votes
3answers
293 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 ...
0
votes
2answers
62 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
1answer
121 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 ...
1
vote
1answer
157 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
808 views

Conjugate Matrix

Let $A$ be a nilpotent square matrix, $J$ be the antidiagonal matrix with 1's on the secondary diagonal (i.e. $J^{2}=E$) and let $B=J A J.$ Suppose we conjugate the matrices $A,B$ by a matrix ...
2
votes
4answers
264 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 ...
1
vote
1answer
67 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 ...
7
votes
0answers
266 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 $g_{12}=g_{21}=g_{13}=g_{31}=1$ ...
4
votes
1answer
134 views

Hankel matrix commuting with a Jacobi matrix

Assume the semi-infinite Hankel matrix $H$ with entries $$H_{i,j}=\alpha_{i+j-1}, \quad i,j\geq1,$$ where $\alpha_{n}\in\mathbb{R}$, is given. It turns out that in some special situations a ...
2
votes
1answer
124 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 ...
5
votes
0answers
136 views

Is there any study about positive definiteness of some matrix space whose matrices don't have to be positive-definite?

--Updated description-- I'm trying to investigate the stability of tensegrity structures, and this question is related to the second order test. Suppose there is a vector space ...
2
votes
1answer
73 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
122 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 ...
15
votes
2answers
1k views

Alternate and symmetric matrices

Greetings to all ! Let me first confess that this question was mentionned to me by Bernard Dacorogna, who doesn't sail on MO. Let $A\in M_{2n}(k)$ be an alternate matrix. Say that $A$ is ...
2
votes
1answer
55 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
103 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
27 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
99 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.
4
votes
1answer
149 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} ...
2
votes
2answers
201 views

quadratic matrix equation

Find all symmetric matrices $X=X^{T}$ such that \begin{align} XDX^{T}=-D \quad (1) \end{align} where $D\ne 0$ is a real diagonal matrix. For example, $X=iI$ satisfies $(1)$. Can you get a ...
9
votes
1answer
347 views

Why does this antisymmetric product factor out a determinant?

Consider a generic $n \times n$ matrix $M$. Define the $(n-1) \times n$ matrix $M_q$ to be $M$ with the $q$th row omitted, and assume that $M_q$ possesses a right inverse, $R_q$: $$R_q = M_q^T (M_q ...
11
votes
3answers
1k views

Diagonalizing a Certain Real and Symmetric Toeplitz Matrix

Consider $0\leq \alpha\leq 1$, and let $A_{\alpha}$ be the Toeplitz $n\times n$ matrix given by $$ A_\alpha := \begin{bmatrix} 1 & \alpha & \alpha^2 & \ldots &\alpha^{n-1} \\\ \alpha ...
10
votes
1answer
165 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
295 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
81 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: $$perm(A)=4^{3m} \#SAT(\phi)$$ ...
13
votes
1answer
462 views

Enumeration of $0-1$ matrices with determinant $1$

Has the number $f(n)$ of $n \times n$, $0{-}1$ matrices whose determinant is $+1$ been enumerated? E.g., for $n{=}2$, there are $f(2)=3$ such matrices: $$ \left( \begin{array}{cc} 1 & 0 \\ 0 ...