The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.

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-3
votes
0answers
35 views

Decomposition of orthogonal matrix into 2 orthogonal matrices [on hold]

Is there anyway to find a decomposition of orthogonal matrix $A$ into 2 orthogonal matrices $P$ and $Q$ such that $A = PQ^T$?
-1
votes
0answers
51 views

About perturbation of spectral radius of a matrix because of diagonal perturbations

Say I have an off-diagonal symmetric $0,1,-1$ matrix $B$ and a set of $2k$ diagonal matrices, $D_{11}, D_{12}, D_{21}, D_{22},..,D_{k1},D_{k2}$. (you can assume the diagonal matrices to be such that ...
-1
votes
0answers
69 views

Inequality for a gradient of a function in Holder space

Let $||X||_{\gamma}=|X|_0+|X|_{\gamma}$, where $|X|_{\gamma}=\underset{x, x'\in \mathbb{R}^3, x \neq x'}{\mathrm{sup}} \frac{|X(x)-X(x')|}{|x-x'|^{\gamma}}$ and $X: \mathbb{R}^N \rightarrow ...
1
vote
0answers
78 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
1answer
60 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
2answers
66 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
2answers
224 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
1answer
77 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
2answers
112 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
1answer
165 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 ...
1
vote
0answers
67 views

How to calculate $det(X^TX)$ efficiently, update one column of X each time [closed]

$X_{1} = (A, b)$, where $X_{1}$ is a $n\times p$ matrix, $A$ is a $n\times (p-1)$ and $b$ is $n\times1$. First calculate $\det(X_{1}^T X_{1})$, then update $b$ with $c$, st. $X_{2} = (A, c)$ and ...
2
votes
1answer
232 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
0answers
46 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
1answer
133 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
0answers
35 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
1answer
53 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
3answers
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
0answers
66 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
0answers
51 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
2answers
202 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
0answers
68 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, ...
18
votes
4answers
712 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
5answers
307 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
1answer
166 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
1answer
219 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
1answer
102 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
0answers
82 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
0answers
57 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
2answers
132 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
0answers
118 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
1answer
121 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
2answers
213 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
97 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
2answers
168 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
0answers
79 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
1answer
85 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
0answers
77 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
1answer
428 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
1answer
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
1answer
155 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
2answers
131 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
0answers
72 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
0answers
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
0answers
76 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
1answer
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
1answer
381 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
1answer
163 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
1answer
106 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
0answers
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 ...
0
votes
0answers
96 views

Question about majorization of eigenvalues after conjugation

Let $A$ and $B$ be $n \times n$ positive semidefinite matrices with eigenvalues $\alpha_1 \ge \alpha_2 \ge \ldots \ge \alpha_n$ and $\beta_1 \ge \beta_2 \ge \ldots \ge \beta_n$ respectively. ...