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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2
votes
2answers
211 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
28 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
14 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
43 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 ...
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)}, ...
1
vote
0answers
57 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
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 ...
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 ...
4
votes
0answers
74 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
0answers
43 views

What kind of matrix is similar to an irreducible matrix? [closed]

For example, we know that the set of positive definite matrix is similar to an irreducible matrix. Therefore, the set of such matrices should include the positive definite matrix cone. What kind of ...
2
votes
1answer
203 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
60 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
64 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 ...
-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 ...
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
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 ...
1
vote
1answer
156 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
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 ...
2
votes
4answers
263 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
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 ...
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.
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
80 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
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} ...
7
votes
0answers
265 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 $g_{12}=g_{21}=g_{13}=g_{31}=1$ ...
2
votes
1answer
66 views

Are there non-isomorphic graphs with rationally orthogonal similar adjacency matrices?

Let $A_G,A_H$ be the adjacency matrices of two non-isomorphic graphs. Let $P$ be orthogonal matrix with rational entries. Is it possible $A_G = P^{-1}A_H P$? Paper gives algorithm for ...
5
votes
2answers
255 views

When are the adjacency matrices of non-isomorphic graphs similar?

From Wikipedia. In linear algebra, two n-by-n matrices A and B are called similar if $$ B = P^{-1} A P$$ for some invertible n-by-n matrix $P$. If $P$ is a permutation matrix, $A$ and $B$ are ...
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 ...
0
votes
0answers
35 views

Changes in singular Values of matrix when adding row

I know that if a column is added to a matrix then the matrix largest signular value increases and the smallest singular value decreases. That is: Given matrix $A \in R^{m \text{x} n}$, $m>n$, and ...
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 ...
3
votes
0answers
80 views

Bound on the ratio of top 2 eigenvalues

Let $P$ be a $(n+1) \times (n+1)$ stochastic matrix such that $P_{ij}=\tau$ if $i \neq j$ and $P_{ii} = (1 - n\tau)$ where $0<\tau < \frac{1}{n+1}$. It is clear that the largest eigenvalue of ...
3
votes
2answers
156 views

What are interesting heuristics of determining how far given matrix is from a singular one?

The condition number and volume of matrix (defined as absolute value of its determinant) are things which come to mind. Is there more? I think that over the years numerical folks (who are faced with ...
5
votes
0answers
87 views

A weak Perron-Frobenius property for sets of positive matrices

A popular form of the Perron-Frobenius theorem states the following result: if $A$ is a $d \times d$ real matrix all of whose entries are positive, then the spectral radius of $A$ is a simple ...
0
votes
0answers
53 views

Is the function below convex?

I have the following function $f(X)=(\sum(gmm^2(AX)-2gmm(AX)gmm(B)))||CX-D||^2$ where gmm is Gussian mixtures defined as $gmm(x)=\sum_{i=1}^{K}\omega_{i}\phi(x|\mu_{i},\Sigma_{i})$, $\omega$ is the ...
0
votes
0answers
90 views

Range of a trace preserving completely positive projection

I'm working in $M_n(\mathbb{C})$, the algebra of complex $n\times n$ matrices. I managed to build a completely positive, trace preserving, star preserving, projection $P$. That is $$\text{Tr}(P(A)) = ...
2
votes
2answers
128 views

How to calculate one Cauchy type determinant

As we know, a Cauchy determinant of size n admits the following explicit formula: $$\det \left(\frac{1}{x _i+y _j}\right) _{1\le i,j \le n}=\frac{\prod _{1\le i < j\le n} (x _j-x _i)(y _j-y ...
1
vote
0answers
149 views

Is there a method to simultaneously block-diagonalize a set of group matrices?

Assume that you are explicitly given the representation matrices of a group. How does one go about finding that common basis which will find the irreducible components of all of them simultaneously? ...
1
vote
0answers
52 views

What are good bounds on ratios of subdeterminants?

Let $A$ be a symmetric matrix and $A_i$ be the matrix obtained from $A$ by dropping the $i^{th}$ row and column. Then what are some good bounds on the value of $\frac{det(A_i)}{det(A)}$ ? Using the ...
5
votes
1answer
249 views

When is a linear combination of permutation matrices unitary?

Question: Let $P_\pi$ denote the matrix representation of permutation $\pi$. Consider a linear combination of all $n \times n$ permutation matrices $$U := \sum_{\pi \in S_n} c_\pi P_\pi$$ where ...
3
votes
0answers
70 views

When is a Hankel matrix invertible?

I would like to have some conditions that render a general Hankel matrix of the form \begin{pmatrix}a_1 & a_2 & a_3 &\cdots & a_n \\ a_2 & a_3 & a_4 & \cdots & a_{n+1} ...
1
vote
2answers
108 views

Matrices congruent to each other via a permutation

Consider the collection of all integer matrices and partition them via an equivalence relation $A\sim B\Leftrightarrow \exists$ a permutation matrix $P$ such that $B=PAP^T$. Is some canonical form ...
1
vote
2answers
137 views

Inverse of a matrix expression

Let $$X_i = \left(I - P\left(I - t_it_i^T\right)\right)^{-1}$$ where $P$ is an $N\times N$ matrix and $t_i$ is a vector of $N$ elements. Is there a way to simplify this expression in order to ...
0
votes
1answer
116 views

$\mathbb{Z}_{2}$ -equivariant vector bundles over manifold of rank-$k$ matrices

Edit: I remove the trivial part of the first version, according to comment of Alex Degtyarev Let $M$ be the manifold of all matrices in $M_{n}(\mathbb{R})$ with fixed rank $0<k<n$. There is ...