**1**

vote

**0**answers

68 views

### What does similar eigenvectors and eigenvalues of two matrices really mean? [on hold]

Empirically I've noticed that diagonally dominant matrix G and it's diagonal version D (diagonal elements of G on the diagonal and all other elements are set to zero) produce similar eigenvalues and ...

**3**

votes

**0**answers

32 views

### Isometric domain of a unital completely positive map with respect to $L_p$-norms

This question can be formulated for general ($\sigma$-finite) von Neumann algebras, but for me it is enough to consider matrix algebras.
So let $M$ be a matrix algebra and $\rho$ a faithful state ...

**2**

votes

**1**answer

49 views

### Regularity of decomposition of matrix-valued function

Here is the problem. Suppose I have a positive definite matrix-valued function $A\colon \mathbb{R}^n\to \mathbb{R}^{n\times n}$. Then we know that there is a matrix-valued function $B\colon ...

**1**

vote

**1**answer

73 views

### Is this matrix positive semi-definite? [closed]

Consider $K$ vectors $x_1,\dots,x_K$ in $\mathbb{R}^N$. Define the $K\times K$ matrix $A$ whose $(i,j)$ entry is given as $$A_{ij}=\exp(-\frac{||x_i-x_j||^2}{2})$$ Is this matrix Positive ...

**0**

votes

**0**answers

35 views

### Composition of upper semi-continuous real valued function with upper semi-continuous matrix valued function

Say that a matrix valued function $A: \mathbb{R} \rightarrow \mathbb{R}^{n \times n}$ is upper semi-continuous at $x_0$ if
$$ \limsup_{x \rightarrow x_0} A(x) \preceq A(x_0), $$
where $\preceq$ ...

**1**

vote

**0**answers

91 views

### 3-regular (cubic) graph with adjacency eigenvalue 1

Suppose $A\in\{0,1\}^{n\times n}$ is the adjacency matrix of a 3-regular (cubic) graph $G=(V,E)$; that is, all $n$ vertices $v\in V$ in the graph have three neighbors.
Is there a nice necessary ...

**5**

votes

**1**answer

65 views

### when does elementwise-log preserve positive-semidefiniteness?

Let $Z$ be a positive semidefinite matrix with nonnegative entries, and define $X=\log(1+Z)$, where the $\log$ is taken entrywise, i.e., $X_{ij}=\log(1+Z_{ij})$. Are there some simple sufficient ...

**4**

votes

**2**answers

76 views

### Integral roots of circulant matrix

When does the circulant matrix have only integral roots?
For example: all roots of the adjacency matrix of the complete graph $K_n$ are integer, which its adjacency matrix is circulant, but in case ...

**2**

votes

**2**answers

314 views

### Proof for the derivative of the determinant of a matrix [closed]

I was looking for theorems that might be helpful in order for some proofs that I have and I came across the following one:
$$\frac{d}{dt} [detA(t)]=detA(t) \cdot tr[A^{-1}(t)\cdot \frac{d}{dt} ...

**2**

votes

**1**answer

80 views

### Any interesting properties of the matrix $M:=(m_{ij})$ with $m_{ij}=min(i,j)$? [closed]

Do you know any interesting properties on the matrix $M(n):=(m_{ij})$ of size $n \times n,$ where $m_{ij}= \text{min}(i,j)$ ? The matrix $M$ enumerates certain combinatorial objects.

**6**

votes

**2**answers

115 views

### Existence and characterization of transitive matrices?

We call a matrix $M \in \mathbb{R}^{d \times d}$ transitive if it satisfies the following:
For any three vectors $u, v, w$ in $\mathbb{R}^d$. If $u^T M v > 0$ and $v^T M w > 0$ then $u^TMw ...

**3**

votes

**1**answer

171 views

### Eigenvalues of the sum of two matrices, where one is $B=\operatorname{diag}(1, 0,\dots,0)$

I know that given two matrices $A$ and $B$, estimating the eigenvalues of $A + B$ by the eigenvalues of $A$ and $B$ is generally a non-easy problem. In particular, there are some results for matrices ...

**3**

votes

**1**answer

386 views

### Comparing norms on tensor products of matrices

Given a Hilbert space $H$, let $S_1(H)$ denote the space of trace-class operators on $H$, with the trace-class norm or Schatten 1-norm. That is
$$ \Vert T \Vert_1 = \sum_{j\geq 1} |s_j| $$
where ...

**0**

votes

**0**answers

50 views

### Can we increase spectral norms of All maximum size square submatrices by orthogonal perturbation?

Let the matrix $A$ consist of $k$ columns from some $n \times n$ orthogonal (unitary) matrix. It is obvious that there is no perturbation of $A$ which
leaves its columns orthonormal,
increases ...

**2**

votes

**1**answer

135 views

### Finding matrices $A$ such that the entries of $A^n$ have specified signs

What techniques are there for ensuring nonnegativity of various entries of matrix powers?
Specific Question: Consider a matrix $A\in SL_2(\mathbb R)$. Let $(A^n)_{i,j}$ denote the $(i,j)$ entry of ...

**0**

votes

**0**answers

38 views

### Column Inner Products vs. Row Inner Products

Given two matrices $A,B\in\mathbb{R}^{n\times r}$ where $A$ has orthogonal columns and $A^TB$ is symmetric, are there any non-trivial interesting relationships / inequalities between the following ...

**1**

vote

**1**answer

55 views

### How to determine an unitary operator involved in an unitary transformation?

Let two real matrices $A$ and $B$ be unitarily equivalent. How to determine (computationally or theoretically) the unitary operator $U$ s.t. $A = UBU^\dagger$? Is it possible for some special class of ...

**28**

votes

**3**answers

2k views

### A curious determinantal inequality

In my study, I come across the following curious inequality, which I do not know a proof yet (so I am asking it here).
Let $A, B$ be $n\times n$ (Hermitian) positive definite matrices. It is very ...

**3**

votes

**0**answers

76 views

### Determining Nullspace Basis such that only one column is deleted or added as row is added or deleted, and remaining columns of basis stay the same

I would like to compute, in MATLAB, the basis Z for the nullspace of an m by n matrix A, such that if one row of A is added (resulting in A_a), the basis for A_a is n-m-1 of the n-m columns of Z, ...

**0**

votes

**0**answers

54 views

### The 2-norm of a positive circulant matrix

Define a circulant matrix $A$ for complex numbers $a_1, a_2, ..., a_n$ as follows:
$$
\text{circ}(a_1,\ldots,a_n)=
\left[ \begin{matrix}
a_1& a_2 & \cdots & a_{n-1} & a_n \\
a_n& ...

**2**

votes

**2**answers

205 views

### If $A>B>0$, can we always find a positive real number $\alpha$, $0<\alpha < 1$ such that $\alpha A \geq B $?

Suppose we have positive-definite matrices $A$, $B$, if $A>B>0$, can we always find a positive real number $\alpha$, $0<\alpha < 1$ such that $ \alpha A \geq B $? If it has, then what ...

**1**

vote

**0**answers

70 views

### Perturbation of eigenvalues of some special matrices

In perturbation theory of linear operators, one major question is how the eigenvalues of a linear operator $A$ change under a small perturbation, $A(x) = A + xP$, with $x\in\mathbb{R}$. For instance, ...

**19**

votes

**4**answers

717 views

### How much redundancy resides in an $n \times n$ orthogonal matrix?

Suppose one has an $n \times n$ orthogonal matrix $M$:
$$
\left(
\begin{array}{ccc}
0.239326 & 0.846726 &
0.475161 \\
0.768893 & 0.13356 &
-0.625272 \\
0.592897 & ...

**7**

votes

**5**answers

319 views

### Packing obtuse vectors in $\mathbb{R}^d$

I came across this attractive theorem:
Theorem. In $\mathbb{R}^d$, there can be at most $d+1$ vectors that
form an obtuse angle with one another.
This was proved1 as a corollary of a lemma about ...

**3**

votes

**1**answer

192 views

### Eigenvectors as continuous functions of matrix - diagonal perturbations

The general question has been treated here, and the response was negative. My question is about more particular perturbations. The counterexamples given in the previous question have variations not ...

**6**

votes

**1**answer

228 views

### Convexity of the product of two exponential matrices

Let $S\subset\mathbb{R}$ be a convex set and $\mathbb{S}^{n}$ be the set of real symmetric matrices of order $n\times n$.
A matrix valued function $\Gamma: S \rightarrow \mathbb{S}^{n}$ is said to ...

**3**

votes

**1**answer

105 views

### Create matrix containing values in [0,1] where sum of all diagonals and anti-diagonals is fixed

The problem I am facing sounds at first glance pretty simple. However, as very often, it seems more complicated than I first assumed:
I want to calculate a matrix $P = (p_{j,k}) \in \mathbb{R}^{n ...

**1**

vote

**0**answers

84 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 ...

**1**

vote

**0**answers

58 views

### FPTAS for approximating the permanent of a matrix

My question concerns approximating permanent of an $n$-by-$n$ matrix.
Several approximation algorithms have been proposed in the literature for this purpose, whose time complexity depend on $n$ and ...

**1**

vote

**2**answers

137 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

**0**answers

123 views

### Bounding the largest Singular value

D is a $n \times n$ diagonal matrix whose diagonal entries lies in $(0,1]$.
B is any $n \times n$ n.n.d. matrix.
What will be the sharpest upper bound on the largest eigenvalue of:
...

**2**

votes

**1**answer

133 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 ...

**2**

votes

**2**answers

217 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

**0**answers

99 views

### Stability of a linear system and spectrum of the product of two matrices

Let us consider an invertible matrix $\mathbf{A}\in GL_d(\mathbb{R})$ such that all its diagonal entries $\mathbf{A}_{ii}=-1 \; \forall \, i$.
My question is the following:
Does it always exists a ...

**3**

votes

**2**answers

169 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 ...

**3**

votes

**0**answers

85 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} ...

**2**

votes

**1**answer

87 views

### Bounds on Hilbert-Schmidt norm of difference of products of matrices

I suspect the following is well-known, but don't know of a reference (and it is not close to the area I normally work in).
I have two sequences of matrices $Q_{1},\ldots,Q_{k}$ and ...

**1**

vote

**0**answers

85 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 ...

**11**

votes

**1**answer

433 views

### A generalization of the Powers-Stormer inequality

The well-known Powers-Stormer inequality says the following: for positive semidefinite operators $A, B$, we have that $\mathrm{Tr}((A - B)(A - B)) \leq \| A^2 - B^2 \|_1$, where $\| \cdot \|_1$ ...

**3**

votes

**1**answer

98 views

### Upper bounds on elements of a matrix

During my research I have come across matrices this type
$$C=B\left(B^T B\right)^{-1}B^T\ ,$$
where $B$ is an $m\times n$ real matrix. If $B^TB$ is not invertible, then $\left(B^T B\right)^{-1}$ ...

**4**

votes

**1**answer

161 views

### Matrix-convexity of inverse of the cofactor matrix

Consider the matrix-valued function $f(A) = \frac{A}{\det(A)}$ on the set of $3\times 3$ positive-definite matrices. Is this function matrix-convex ? (i.e., is $tf(A) + (1-t)f(B) - f(tA+(1-t)B)$ ...

**3**

votes

**2**answers

132 views

### Positive definiteness of infinite tridiagonal matrices

I am interested in the following problem: I have an infinite symmetric tridiagonal matrix
$$
A=
\begin{bmatrix}
a_1 & b_1 & & & \\
b_1 & a_2 & b_2 & & ...

**1**

vote

**0**answers

73 views

### Algorithms to compute the rank of a parametrized matrix [closed]

Motivated by my question on Mathematics StackExchange and by a question by Anirbit on the same site, I ask for some references on the problem of rank computation for a parametrized matrix. References ...

**0**

votes

**0**answers

52 views

### Dual cone of a set of particular semidefinite cones

Let $X$ be a matrix variable
$$X=\begin{pmatrix} x_1 & x_2 & x_3\\ x_2 & x_4 & x_5\\x_3 & x_5 & x_6\end{pmatrix},$$
define the cone as
...

**1**

vote

**0**answers

82 views

### Unitary transformation of a Hermitian indefinite pencil to a real non-symmetric pencil

Given a Hermitian indefinite pencil $(A-\lambda B)$ where both $A=A^H$ and $B=B^H \in \mathbb{C}^{n\times n}$ are possibly indefinite, it is straightforward to show that the eigenvalues are either ...

**3**

votes

**1**answer

119 views

### Normalizing a matrix via triangular transformations

Consider the following $n\times n$ real-valued matrix:
$$
A=\begin{bmatrix}
\alpha_1 & \beta_1 & 0 &\cdots & 0 &\gamma_n\\
\gamma_1 & \alpha_2 & \beta_2 & \cdots & ...

**3**

votes

**1**answer

488 views

### How to calculate the square root of matrix $A+B$ perturbatively?

$A=diag\{\lambda_1,...,\lambda_n\}$ and $\lambda_i>0$, $B$ is a positive definite symmetric matrix and $max\{B_{ij} \}\ll min\{\lambda_i\}$
Note that the perturbative calculation of square root ...

**5**

votes

**1**answer

167 views

### How many cospectral graphs available for a given number of nodes?

Two graphs are said to be cospectral if they have same eigenvalues wrt adjacency matrix, Normalised or Signless laplacian matrix. How many graphs has cospectral mates for a given number of nodes? We ...

**-1**

votes

**1**answer

112 views

### How to show the square root function of a positive semidefinite matrix is differentiable? [closed]

How to show the square root function of a positive semidefinite matrix is differentiable?
In this context PSD means symmetric PSD.

**1**

vote

**0**answers

80 views

### Distribution of a signal covariance matrix

A common estimation problem in signal processing assumes the following signal model
\begin{equation}
\mathbf{r} = \sum_{i=1}^{Q}\alpha_i\mathbf{s}\left(w_i\right)+\mathbf{n}
\end{equation}
where ...