**1**

vote

**0**answers

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

**7**answers

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

**2**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**4**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**3**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**4**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**3**answers

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

**1**answer

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

**5**answers

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

**0**answers

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

**1**answer

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 ...