**2**

votes

**0**answers

44 views

### Unitary factor in polar decompositions

Let $A, B$ be $n$-square (Hermitian) positive definite matrices. Let $AB=U|AB|$ be the polar decomposition of $AB$. So $U$ is unitary (called the unitary factor of $AB$). What is the optimal constant ...

**-2**

votes

**0**answers

46 views

### An analytic characterization of eigenvalues of a Hermitian matrix [on hold]

[..the following is trying to understand a certain argument of Terence Tao in a lecture notes of his..]
If $A$ is a Hermitian matrix with eigenvalues, eigenvectors $\{ \lambda_i , e_i \}_{i=1}^n$. ...

**5**

votes

**0**answers

59 views

### Powers of traces, integrals over spheres and class functions

I asked this on math.StackExchange a while back but got no answers. I hope I'll be forgiven for the double post.
Let $V$ be a complex vector space of dimension $\operatorname{dim}_{\mathbb C} V = n$, ...

**0**

votes

**0**answers

42 views

### Matrix equation

Let $A$ be $k\times n$ matrix i.e., $A=(a_{1},\ldots, a_{n})$ where $a_{j} \in \mathbb{R}^{k}$, $rank(A)=k$ and $1\leq k \leq n$. Let $q=(q_{1},\ldots, q_{n})\in\mathbb{R}^{n}$ be such that ...

**2**

votes

**1**answer

72 views

### Do the sequences with divergent associated $\zeta$-function form a vector space?

Let $V$ be the set of sequences $a \in\mathbb{R}^\mathbb{N}$ such that $\lim_{n\to\infty} a_n = 0$. The set $V$ can be seen as a real vector space, with pointwise addition and scalar multiplication.
...

**0**

votes

**0**answers

16 views

### Solution to a system of linear equations containing some inequalities [on hold]

I have a system of equations as follows:
$a_{11}x_1 + a_{12}x_2 + a_{13}x_3 = b_1$
$a_{21}x_1 + a_{22}x_2 + a_{23}x_3 = b_1$
$a_{31}x_1 + a_{32}x_2 + a_{33}x_3 < b_1$
$a_{41}x_1 ...

**-4**

votes

**0**answers

28 views

### Find the vector component of vector u orthogonal to vector a [on hold]

I have vector u = (-2, 3, 1) and vector a = (-2, 2, 2). How do I find the vector component of u orthogonal to a?
I've done the cross product and I get (-4,-2,-2), but I am assuming that this is also ...

**-3**

votes

**0**answers

57 views

### About diagonal entries of the graph Laplacian

[..in the following you can assume its a regular graph if necessary..]
Is anything special known about them?
Are they characterized in any other way?
Is the largest diagonal entry in any power of ...

**1**

vote

**0**answers

77 views

### Can anyone help me deduce a matrix inequality?

The following lemma is taken from references firstly.
Lemma 1 [1-2] Given matrices $Q=Q^{T} , F, M$ and $N$ of appropriate dimensions, then $$Q+MFN+N^{T}F^{T}M^{T}<0$$
for all $F$ satisfying ...

**0**

votes

**0**answers

104 views

### Alternate proof of Schur orthogonality relations [migrated]

I am trying to find an alternate proof for Schur orthogonality relations along the following lines.
Let $G$ be a finite group, with irreducible representations $V_1$, $V_2$, $\cdots$, $V_d$.
Let $V$ ...

**6**

votes

**0**answers

68 views

### In what sense is the Bayesian posterior mean a “convex combination”?

I asked this on math.stackexchange with no response, I'm hoping someone here might have something.
Suppose I want to estimate $x \in \mathbb{R}^n$ from two signals with zero mean, normally ...

**1**

vote

**0**answers

91 views

### Infinite matrices with a finite number of non-zero values on each row

The little bit of literature on infinite matrices I've been able to find studies a general setting in which the theory is hindered by constantly having to worry about whether or not various infinite ...

**1**

vote

**2**answers

267 views

### How to prove that a kernel is positive definite?

For example, how to prove
$\forall(x,y)\in R^N\times R^N,K(x,y) = \displaystyle\frac{1}{1+\frac{||x - y||^2}{{\sigma}^2}}\\$
where $\sigma > 0$ is a parameter, is positive definite? I have tried to ...

**-3**

votes

**3**answers

268 views

### Determinant of matrix from set {-1, 1}

Let $A \in \mathbb{R}^{11 \times 11}$ and it's elements are form set $\{ -1,1 \}$. $\mathbb{P}(-1) = \mathbb{P}(1) = 0.5$. What is a probability to get such a matrix, that $\det A > 4000$?
I have ...

**3**

votes

**0**answers

40 views

### Characterizing space that preserves positive-definiteness property

Given a symmetric positive-definite matrix $\Sigma$, consider the space $\mathcal{D}$ of diagonal matrices such that $\forall D\in\mathcal{D}$, the matrix $\Sigma-D\Sigma^{-1}D$ is positive definite. ...

**2**

votes

**2**answers

154 views

### When does a Vandermonde-like matrix have full rank

I have a matrix which is similar to Vandermonde matrix except that the entries are monomials of degree $d$ polynomial in 2 variables. Each row has the following form:
$X_{i}= [1, x_{i}, y_{i}, ...

**1**

vote

**0**answers

33 views

### Distributing partially known data between n parties

Assume that $n = 2r+1$. There are $n$ elements $a_1,a_2,\ldots,a_n$ from a finite field $\mathcal{F}$, and $n$ parties. Each party knows the values of at least $r+1$ elements out of those $n$ ...

**0**

votes

**0**answers

37 views

### RIP, NSP and Spark of A Matrix [closed]

Definition: A matrix A∈Cm×N with $m<N$ satisfies the Null Space Property (NSP) of order s if
$∥x_{S}∥_{1}<∥x_{S}^{c}∥_{1},$ $∀x∈kerA$\{0}$,∀S⊆{1,…,N} s.t. |S|≤s $,
where $x_{S}=(x_{i})_{i} ∈S$.
...

**3**

votes

**0**answers

27 views

### Counting cosets of matrices of determinant > 1 under the action of a congruence subgroup

I tried asking this on math exchange, but no luck, so thought I'd try here.
Let $M_2(m,\mathbb{Z}) $ be the $2\times 2$ matrices with integer entries and determinant $m$. Let $\Gamma^0(N)$ be the ...

**0**

votes

**0**answers

18 views

### Composition of Lossless Systems from Delay and Mixing regarding junction admittance

Given $m_1, \dots m_N \in \mathbb{N}$ and matrix $\mathbf{A} \in \mathbb{C}^{N\times N}$.
Let us define a diagonal matrix $\mathbf{D}(z) = diag(z^{-m_1}, \dots, z^{-m_N})$ with $z\in\mathbb{C}$. ...

**6**

votes

**2**answers

402 views

### About Sylvester's determinant

If $A$ is any $n \times m$ matrix and $B$ is any $m \times n$ matrix then one familiar form of the Sylvester's identity is $\det(I + AB) = \det(I + BA)$.
Now somehow curiously this above identity ...

**4**

votes

**1**answer

145 views

### Dirichlet Characters as Eigenvectors

This was asked in Math Stackexchange here but generated no comments or answers. I have slightly edited the original question with the comment in the fourth paragraph and the explicit matrix example at ...

**2**

votes

**0**answers

75 views

### Copositivity under tensor products

Is there any symmetric real matrix $A$ such that $A^{\otimes n}$ is copositive for all positive integers $n$, but such that A is neither positive semidefinite nor has just non-negative entries?
...

**3**

votes

**0**answers

108 views

### Commutative decomposition for full-rank $A$ and low-rank $B$ matrices that do not commute

1. Motivation
Consider symmetric matrices $A,B\in\mathbb{R}^{n\times n}$, and let $A$ be full-rank and $B$ be low-rank. The simultaneous block-diagonalization, defined as the following
...

**1**

vote

**0**answers

85 views

### Pre- and post-multiplication by diagonal matrices [closed]

Let $\mathbf{1}$ denote an $n\times 1$ vector with all entries equal to 1. Given an $n\times n$ matrix A with strictly positive entries, and non-negative diagonal matrices $D_1$ and $D_2,$ evaluate ...

**13**

votes

**6**answers

682 views

### Does the linear automorphism group determine the vector space?

I was recently thinking about what it means to put structure on a set. It seems to me that, in my area (representation theory), the two main ways of imposing structure on a set $X$ are:
...

**1**

vote

**1**answer

70 views

### Nonlinear system of equations whereas most of the equations are linear. How to minimise operation?

Let us say we have a n * n system of equations like KU=F where K is a n*n matrix and U and F are n*1 vectors. K and F are defined and the final goal is to find U values.
K is a sparse banded matrix ...

**11**

votes

**0**answers

221 views

### Ratio of entries of A and log A where A is a triangular matrix

Consider triangular matrices $A = \left( {a(n,k)} \right)$ of arbitrary order with $a(n,k) = 0$ if $n + k$ is odd and $a(n,n - 2k) = \frac{{n!}}{{k!(n - 2k)!}}\frac{{(m + n - k - 1)!}}{{(m + n - ...

**4**

votes

**0**answers

106 views

### minimal polynomial for a graph

I wonder if there is any result relating the degree $d$ of the minimal polynomial of a directed finite graph to any of its topological features - such as its diameter, or any other similar 'natural' ...

**2**

votes

**1**answer

130 views

### Let $R$ be a $M\times N$ matrix with rational entries, Is $|(R\mathbb{Z}^N)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M)/\mathbb{Z}^N|$?

Let $R$ be a $M\times N$ matrix with rational entries, $R\mathbb{Z}^N$ be the image of $\mathbb{Z}^N$ under R.
Consider a equivalent relation on $R\mathbb{Z}^N$ defined by
$a\sim b$ if $a-b\in ...

**5**

votes

**2**answers

252 views

### Partial inverse of a matrix - or does it have its own name?

In my calculations I need to use something which is "between" a matrix and its inverse. That is, I invert only some dimensions. I am interested if it has an established name.
That is, a matrix (here ...

**23**

votes

**2**answers

1k views

### When exactly and why matrix multiplication became a part of undergraduate curriculum?

The story about Heisenberg inventing matrices and matrix multiplication in 1925 is very well known and well documented. Few weeks later Born and Jordan picked this and recognized the matrix ...

**1**

vote

**0**answers

49 views

### Decomposition of non-singular matrix [closed]

Is there any way to show that a non-singular matrix A can be partitioned as follows:
\begin{eqnarray*}
A&=&\left[
\begin{array}{cc}
\underset{\left( k\times k_{1}\right) ...

**38**

votes

**1**answer

724 views

### Determinant of a determinant

Consider an $mn \times mn$ matrix over a commutative ring $A$, divided into $n \times n$ blocks that commute pairwise. One can pretend that each of the $m^2$ blocks is a number and apply the $m ...

**0**

votes

**0**answers

135 views

### Problem regarding orthogonal vectors

Suppose $C_{1},C_{2},...,C_{n}$ are $0-1$ vectors of length $m$. Given $C_{i} \in \{0,1\}^{m}$ with $C_{i}=x_{i1}x_{i2}...x_{im}$ we say $C_{i}'=x_{i1}'x_{i2}'...x_{im}'$ is a subvector of $C_{i}'$ if ...

**5**

votes

**0**answers

156 views

### Product $PVPVP$ is elementwise nonnegative?

Let $P\in \mathbb{R}^{n\times n}$ be the inverse of a positive definite M-matrix and $V\in \mathbb{R}^{n\times n}$ be any diagonal matrix. Prove (or disprove) that $PVPVP$ is elementwise nonnegative.
...

**5**

votes

**3**answers

102 views

### Minimize distance between centroids of subsets of points

In a n-dimensional space, I want to divide a set of m points into v (non-empty) subsets.
I want to minimize the sum of the pairwise Euclidean distances between the centroids of the resulting subsets.
...

**5**

votes

**1**answer

181 views

### Anti-bidiagonal matrix with main anti-diagonal {1,2,3,…} and first sub-anti-diagonal {-1,-2,-3,…} has eigenvalues lambda={1,-2,3,-4,…}

Consider the anti-bidiagonal matrix $B_6\in\mathbb{R}^{6\times 6}$, defined along its anti-diagonals as follows
$$
B_6=\begin{bmatrix} & & & & & 6\\
& & & ...

**5**

votes

**1**answer

479 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

174 views

### Do singular values dominate eigenvalues?

Suppose $A$ is an $n \times n$ complex matrix with singular values $s_1 \ge s_2 \ge \cdots \ge s_n$ and eigenvalues $(\lambda_i)_{i=1}^{n}$ arranged so that $|\lambda_1| \ge |\lambda_2| \ge \cdots ...

**1**

vote

**2**answers

154 views

### Question on Posets and open sets [closed]

i'm sorry if my question is really trivial but this one is really bugging me out..
So let's have a partially ordered set $I$ with the topology in which the open sets are the increasing ones: $i\in U$ ...

**7**

votes

**2**answers

183 views

### Are sums of 0-1 Pareto efficient vectors Pareto efficient?

Does there exist $m,n\ge1$, an $m \times n$ matrix $A$, and a vector $x \in \mathbb{R}^n$ such that:
The entries of $A$ are $\in \{0, 1\}$.
For all pairs of columns $u, v$ of $A$ the entries of $u - ...

**0**

votes

**0**answers

37 views

### Questions about some special tensor transformation

Suppose tensor $U_{i\alpha\beta}$ with dimension $M*N*N$ satisfy following condition:
$$U_{i\beta\alpha}=W^1_{\alpha\alpha'}W^2_{\beta\beta'}U_{i\alpha'\beta'}$$
where $W^1$ and $W^2$ are $N*N$ ...

**1**

vote

**1**answer

149 views

### Is the notation ${}^t g$ for the transpose of a linear transformation intended to be suggestive?

The notation ${}^t g$ for the transpose of a linear transformation is, in my view, quite unusual: otherwise (at least in many areas of math), one almost never sees subscripts or superscripts appearing ...

**2**

votes

**1**answer

104 views

### Which nonnegative matrices have exact nonnegative matrix factors of smaller dimensionality?

The nonnegative matrix
$V = \left( \begin{array}{cc}
1 & 1 \\
1 & 1 \end{array} \right)$
has nonnegative matrix factors $W = \left( \begin{array}{c} 1 \\ 1 \end{array} \right)$ and $H = ...

**2**

votes

**0**answers

58 views

### Is there an efficient way to compute the “complete subset regression”?

Background: Let $X \in \mathbb{R}^{N\times K}$ and $y \in \mathbb{R}^{N\times 1}$ be data for a regression problem. The aim is to find $\beta \in \mathbb{R}^{K\times 1}$ such that $X\beta \approx y$ ...

**2**

votes

**1**answer

79 views

### Linearly constrained eigenvalue problem

Suppose I'd like to:
\begin{align}
\mathop{\text{min}}_\mathbf{x} && \mathbf{x}^T\mathbf{A}\mathbf{x} \\
\text{subject to:} && \mathbf{x}^T \mathbf{M} \mathbf{x} = 1\\
&& ...

**1**

vote

**0**answers

20 views

### How to restructure adjacency matrix $A$ from shortest distance matrix $B$ in Network topology inference

An undirected graph with $n$ nodes could be referred to as an adjacency matrix $A$. $A=[a_{ij}]_{n×n}$ with $a_{ij}=a_{ji}=1$ standing for there being an edge between node $i$ and node $j$, and no ...

**4**

votes

**1**answer

106 views

### Writing a complex orthogonal matrix as a conjugation by real orthogonal matrices

Let $Q\in O(n,\mathbb C)$ be a complex orthogonal matrix. I would like to know if $Q$ can always be written as $Q = T^{-1}ST$, where $T\in O(n,\mathbb R)\subset O(n,\mathbb C)$ and $S$ belongs to some ...

**1**

vote

**1**answer

55 views

### Looking for algorithms based on sorting [closed]

i am looking for algorithms which use sorting in low-dimensional space like $R$ and how they are generalized for higher-dimensional spaces like $R^2$ where there is no sorting possible. (i.e. numbers ...