**0**

votes

**0**answers

18 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. ...

**0**

votes

**0**answers

21 views

### SVD of sum of a symmetric matrix and a banded matrix [on hold]

The SVD of the sum of a symmetric square matrix and an appropriately sized identity matrix can be written as following:
$A^TA + \lambda I = V(\Sigma + \lambda I)V^T$
where $\Sigma$ contains the ...

**4**

votes

**0**answers

173 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 $g_{12}=g_{21}=g_{13}=g_{31}=1$ ...

**1**

vote

**1**answer

84 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

140 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

179 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

335 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 ...

**9**

votes

**1**answer

133 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

270 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

78 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

**0**answers

29 views

### How to Solve this Matrix Question [on hold]

So I've been given this question which is one two by two matrix, and I believe multipled by another two by two matrix which = the matrix
[ 1 0
0 1]
The ...

**13**

votes

**1**answer

414 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 ...

**2**

votes

**1**answer

60 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 ...

**4**

votes

**1**answer

104 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 ...

**0**

votes

**0**answers

169 views

### Theorems about matrices with entries from $0,1,-1$? [closed]

Consider matrices which are of the form $\left [ \begin{matrix}
0 && A\\
A^T && 0 \\
...

**5**

votes

**2**answers

214 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 ...

**0**

votes

**0**answers

44 views

### Determinant of a Vandermonde matrix of roots of monic polynomial with integer coefficients [migrated]

Let $p(x)=\sum_{i=1}^n a_ix^i$ with $a_i$ an integer for all $i$ and $a_n=1$ such that $p(x)$ has only real roots, and let $\lambda_1,\ldots,\lambda_n$ be the $n$ roots of this polynomial. Then the ...

**3**

votes

**0**answers

70 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 ...

**2**

votes

**1**answer

292 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 ...

**17**

votes

**1**answer

1k views

### Smallest eigenvalue of a tricky random matrix

While experimenting with positive-definite functions, I was led to the following:
Let $n$ be a positive integer, and let $x_1,\ldots,x_n$ be sampled from a zero-mean, unit variance gaussian. Consider ...

**0**

votes

**0**answers

30 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 ...

**3**

votes

**2**answers

151 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

**0**answers

83 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

**0**answers

51 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

**0**answers

78 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

**2**answers

120 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

**0**answers

142 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?
...

**7**

votes

**1**answer

417 views

### Determine if a matrix is unimodular

Is deciding if an integer square matrix has determinant $\pm 1$ faster that calculating the determinant of the matrix?

**1**

vote

**0**answers

51 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 ...

**1**

vote

**1**answer

154 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 ...

**5**

votes

**1**answer

239 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 ...

**-1**

votes

**2**answers

2k views

### When does a matrix equation have a solution? [closed]

Given a matrix equation $Ax=b$ where $A$ is a matrix and $b$ is a column vector, what is a condition that would ensure that there is a column vector $x$ that satisfies the equation?
Assume the ...

**1**

vote

**1**answer

763 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 ...

**3**

votes

**0**answers

68 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

**2**answers

134 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 ...

**1**

vote

**2**answers

105 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 ...

**0**

votes

**1**answer

113 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 ...

**4**

votes

**1**answer

126 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

113 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 ...

**0**

votes

**2**answers

478 views

### Similarity about unitary matrices

Suppose $G_1, \ldots, G_k$ are unitary, Hermitian, and anti-commuting matrices, and assume the same for $F_1, \ldots, F_k$. Suppose these matrices are similar, i.e. there exists $T \in ...

**5**

votes

**2**answers

662 views

### Proving that the kernel of this matrix is of dimension 2

(Edit : see at the bottom of the question for an additional surprising possible hint.)
Using a computational software program, I found that the kernel of the following matrix is of dimension 2 when ...

**-4**

votes

**2**answers

237 views

### Representing quaternions as matrices [closed]

Assume F is a field of characteristic different than 2. Let a, b be invertible elements in F, and let A(a,b) be the generalised quaternions. Using the Artin–Wedderburn theorem, there is a ...

**1**

vote

**0**answers

42 views

### how to find coordinate of unknown point given the distance against N known points [migrated]

I am meeting with a problem, say I have already know the coordinates of N points (a1,a2,a3....) in 3D space. And I have a new point, say x. I only know the distances from x to the known N points. Is ...

**3**

votes

**1**answer

83 views

### Probe permutationally matrix extreme properties

Suppose given $M\in\{0,1\}^{n\times n}$ of rank $r$.
Assume that changing even a single $1$ to $0$ in $M$ raises rank. Does it follow that $M$ is permutationally equivalent to a block diagonal ...

**3**

votes

**1**answer

447 views

### Why are 1 and -1 eigenvalues of this matrix?

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix $\mathbf{A}$.
First, let's define two matrices:
...

**3**

votes

**1**answer

219 views

### Finite codimensional subvector space of $C^{*}$ algebras which contains no invertible elements

Assume that $A$ is a unital $C^{*}$ algebra. Is there a subvector space $Y\subset A$ of finite codimension which does not contain any invertible element?
Let $n(A)$ be the infimum of such ...

**1**

vote

**0**answers

45 views

### Spectral radius of a column stochastic matrix perturbed by a rank-1 matrix

$P\in \mathbb{R}^{n\times n}$ is an irreducible column stochastic matrix. $P$ is also diagonally dominant. $w \in \mathbb{R}^{n} $ is a strictly positive vector satisfying $w^T \mathbf{1} = 1$ where ...

**5**

votes

**0**answers

144 views

### Characterizing matrices with rank constraint

Given matrix $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M,b]=\{Q\in\Bbb R_{\geq0,\leq b}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad ...

**3**

votes

**1**answer

81 views

### Maximizing Frobenius Norm of Commutator (an opposite Procrustes problem)

I was wondering if anybody has any suggestions on the following problem:
Let $S$ be an $n\times n$ positive definite symmetric matrix. I wish to find an $n\times n$ orthogonal matrix $R$ which ...