3
votes
1answer
115 views

Alike looking matrices imply convergence of eigenvalues?

This is a question about convergence of eigenvalues which essentially came up in studying the spectrum of St.-Liouville operators. We want to look at matrices that agree in most of their entries and ...
3
votes
2answers
84 views

Norm of triangular truncation operator on rank deficient matrices

Let $T_{n\times n}$ be a triangular truncation matrix, i.e. $$T_{i,j}=\begin{cases}1 & i\ge j\\ 0 & i<j \end{cases}$$ It is known that for arbitrary $A_{n\times n}$ $$\|T\circ ...
0
votes
1answer
70 views

Solution of infinite dimension linear system

Suppose that ${a_n}$ and $b_n$ is decreasing sequence such that $a_0=A$, $lim_{n->\infty}a_n=0$ and $b_0=B$, $lim_{n->\infty}b_n=0$. For fix n, we can construct n dimension linear equation ...
4
votes
1answer
57 views

An extension of the Golden Thompson inequality

For three symmetric positive-semidefinite matrices $A,B,C$ I am trying to figure out if the following inequality holds, at least in some cases: $$tr(Ae^{B+C})≤tr(Ae^Be^C)$$ Note that if $A=I$ then ...
5
votes
7answers
479 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 ...
3
votes
0answers
130 views

Method to Generate Random Mutually Orthogonal Unitary Matrices

The title says it all, I'd like to know if such a method exists to generate random mutually orthogonal unitary matrices. If so, any supplementary references would be very much obliged. Thanks again!
1
vote
0answers
69 views

spectrum of a special class of tridiagonal matrices

Consider a real and symmetric tridiagonal matrix with zero diagonals and where subdiagonals and superdiagonals are equal to 1 except the (1,2)-th component being equal to $a$, i.e., ...
4
votes
2answers
202 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 ...
5
votes
1answer
245 views

Is the p-norm of a matrix strictly log-convex?

Let $A$ be a $n\times n$-matrix. We let $\|A\|_p$ denote the norm of $A$ when considered as a linear operator on $\ell^p(\{1,2,\ldots,n\})$, that is, $$ \|A\|_p = \sup_{x\neq ...
1
vote
0answers
68 views

Inverse of matrix of generalised harmonic numbers

For $s=0,1,\dots$ and $n=1,2,\dots$, denote $r_{n,s}=\sum_{k=1}^n k^s$. It is well-known that $r_{n,s}$ are polynomials in $n$ with leading term $\frac{1}{s+1}n^{s+1}$. Let $R_{n,s}$ be the ...
5
votes
0answers
246 views

Symmetric matrices with $\rho(A)\gg\|A\|_\infty$

For a symmetric real matrix $A$, denote by $\rho(A)$ the spectral radius of $A$, and by $\sigma(A)$ the largest absolute row sum of $A$; that is, $\sigma(A)=\max_i \sum_j |a_{ij}|$, where $a_{ij}$ are ...
2
votes
0answers
150 views

Norm bound of a complex resolvent

A well known result by Varah states that if $A$ is a strictly diagonally dominant matrix of dimension $n$, then $\|A^{-1}\|_{\infty} \le \max_i\frac{1}{|a_{kk}|-\sum_{j \neq k}|a_{kj}|}$, where the ...
0
votes
0answers
55 views

Property of quasipositive matrices

I saw this theorem stated in a paper without proof and I have difficulty proving it. If $A$ is an $n\times n$ matrix with non-negative off-diagonal entries, let $s(A)$ be the real eigenvalue such ...
3
votes
1answer
190 views

When is this matrix singular?

Consider matrix $A$ with $(j,k)$′th entry $A_{j,k}=\sin(\omega_j t_k+\phi_j),\,\forall j,k\in\{1,2,...,n\}$, where $\omega_j,t_k,\phi_j\in \mathbf R$. 1) For $t_k=k$, what is the condition on ...
2
votes
1answer
135 views

derivative of sum of singular values

can someone point me to the direction how to calculate the derivatives of a sum of singular values of a matrix? I am trying to minimize $$\min_A \parallel A \parallel_*+ \cdots $$ where $\parallel A ...
1
vote
0answers
163 views

Incoherence of the row/column span

Due to V.Chandrasekaran., et al‎ (p.11) : In general for any $k$-dimensional subspace of $A_{n×n}$ we have that: $$\sqrt{(k/n)} \leq incoherence(A)\leq 1$$ where the lower bound is achieved (for ...
0
votes
0answers
206 views

Proving that a certain matrix inverse is always positive definite

Take any positive definite Hermitian matrix $X$. Put all values of $X$ equal to 0 except for the values along the center $2K+1$ diagonals which are kept untouched. Denote the new matrix by $Y$. Let ...
0
votes
1answer
177 views

Nonnegative Matrix

Let $A=E+\sqrt{-1}B$, where $E=diag\{0,1,\cdots,1\}$, $B$ is a real symmetric matrix. Let $A^*$ denote the adjoint matrix of $A$, i.e. $AA^*=\det A\cdot I$. I hope the real part of adjoint matrix ...
1
vote
1answer
134 views

Weighted Spectral l-2 norms arising from matrix inner products

The spectral $l^2$ norm of a complex matrix is given by: $\|A\|= \left( \sum_{k=0}^{N-1} s_k(A)^2 \right)^{1/2}$ where $s_k(A)$ are the singular values of $A$ ordered so as to be non decreasing in ...
3
votes
1answer
90 views

Decomposition of a semi-definite matrix into sums of sparse semi-definite matrices

I'll first provide the background. Let $x\in\mathbb{R}^N$ be decomposed into $n$ non-overlapping blocks of variables $x^{(1)},\ldots,x^{(n)}$. We say that $f:\mathbb{R}^N\rightarrow\mathbb{R}$ is ...
2
votes
0answers
106 views

A - B is semidefinite, what the relationship about their eigenvalues? [closed]

$A, B$ are two symmetric matrices, if $ A-B $ is semidefinite (i.e.$ A - B \geq 0$), if we rearrange the eigenvalues of two matrices, $\lambda_1 (A) \geq \lambda_2 (A) \geq ... \geq \lambda_n (A)$, ...
5
votes
3answers
359 views

Optimization problem on trace of rotated positive definite matrices

Given two $n \times n$ symmetric positive definite matrices $A$ and $B$, I am interested in solving the following optimization problem over $n \times n$ unitary matrices $R$: $$ \mathrm{arg}\max_R ...
2
votes
1answer
93 views

Dimension independent computational complexity of singular value decomposition

Suppose $X$ is a $m \times n$ real matrix, which has only $k$ number of nonzero elements ($k \ll mn$). Given a vector $y$, the sparsity of $X$ allows $X y$ to be computed in $O(k)$ time which is ...
6
votes
1answer
330 views

The singular values of the Hilbert matrix

The $n\times n$ Hilbert matrix $H$ is defined as $H_{ij} = \frac{1}{i+j-1}, \qquad 1\leq i,j\leq n$ What is known about the singular values $\sigma_1\geq\ldots\geq \sigma_n$ of $H$? For example, ...
1
vote
0answers
84 views

Copositivity in matrix pencils

Given two square symmetric matrices $A,B$ of the same order, the matrix pencil $P(A,B)$ is the set of linear combinations of $A$ and $B$. Finsler's theorem gives an elegant criterion for $P(A,B)$ to ...
8
votes
1answer
214 views

Inequalities for Hadamard products of complex symmetric matrices

Consider a complex symmetric matrix $$ C= C_R + i C_I $$ with $C_R,C_I \in \text{Mat}_{n\times n}(\mathbb R)$ symmetric, and assume that the eigenvalues of $C_R$ are all strictly positive. Then, $C$ ...
1
vote
0answers
80 views

Matrices with a common fischer basis

Let $A$ be a real symmetric $n\times n$ matrix normalised such that $Tr[A]=1$. Define 'fischer basis' as the basis in which all diagonal elements are equal to $\frac{1}{n}$. The motivation for this ...
0
votes
1answer
606 views

How to compute difference between 2 similarity matrices?

Hello, I have two n*n correlation matrices with values ranging between -1,1. (2 correlation matrix because I have the same n terms under 2 different conditions) I then transformed the correlation into ...
4
votes
2answers
249 views

system of homogeneous matrix equations

Edit: Maybe I should make it clear that by a solution I mean a pair of matrices $(A,B)$ such that the identity below holds for any complex numbers $x,y$. One of my friend asked me the following ...
16
votes
1answer
623 views

2x2 subdeterminants of a matrix

If N>2, it is well known that if two invertible NxN matrices A and B have the same determinants of any 2x2 corresponding submatrices, then A=B or A=-B. Given then all these 2x2 determinants of an ...
9
votes
2answers
845 views

On the Positive Definiteness of a Linear Combination of Matrices

In my work in PDE, the following problem in linear algebra came up. Any help in this direction is appreciated. QUESTION: Let $m,n\in\mathbb{N}$ and let $A_1,\ldots, A_m\in M_n(\mathbb{R})$ be real, ...
2
votes
1answer
221 views

Asymptotic Behavior of Non-Analytic Function of the Eigenvalues

Hello, Let $A_n = (a_{k-j};\;k,j = 0,1,\ldots,n-1)$ be a sequence of $n\times n$ Toeplitz matrices, with eigenvalues $(\lambda_{n,i};\;i = 0,1,\ldots,n-1)$. If $A_n$ were a sequence of Hermitian ...
3
votes
1answer
254 views

Singular values of the sum of A and A^T

Dear all, As a part of my research, I need to achieve a lower bound to the smallest singular value, $\sigma_{n}\(A+A^{T}\)$ for a stochastic $A$ (as a function of the singular values of $A$), which ...
3
votes
2answers
114 views

a monotone relation for s-numbers

Assume $A, B$ are self-ajoint compact operators. Is it true that $\|A+iB\|\le \|2A+iB\|$? Do we have a stronger inequality $\prod_{k=1}^ns_k(A+iB)\le \prod_{k=1}^ns_k(2A+iB)$ or even stronger one ...
5
votes
1answer
301 views

A matrix inequality involving the Hilbert-Schmidt norm

This question comes from a problem in PDEs on which I'm currently working. Let $a$ be a $3\times 3$ matrix, real symmetric and positive definite. Denote with $\|a\|^2 _ 2=\sum a_{ij}^2$ the square of ...
1
vote
1answer
356 views

power of a block triangular matrix

I have a matrice in the form : $$M = \begin{pmatrix} A & 0 & 0 \\\ B & A & 0 \\\ C & D & A \end{pmatrix} $$ where $A,B,C,D$ are diagonalizable square matrice and I want to ...
5
votes
2answers
248 views

Simultaneous maximization of two Generalized Rayleigh Ritz Ratios

Consider hermitian positive semi-definite matrices $A_1$ and $A_2$. Consider also positive definite matrices $B_1$ and $B_2$. I want to maximize the minimum of the two Generalized Rayleigh Ritz ratios ...
0
votes
1answer
279 views

Simultaneous Jordanization

Hello everyone I would like to have a detailed reference to the statement bellow: Let $A,B\in \mathbb{R}^{n\times n}$ such that $AB=BA$. Suppose $A$ has real eigenvalues only and $B$ is ...
1
vote
3answers
376 views

Number of parameters needed to specify a Hermitian matrix of rank r.

Hi, i have two questions that seem to bother me lately. Maybe you could help me and/or point me in any related literature. 1) Assume hermitian matrix $H \in \mathcal{C}^{n \times n}$ that has rank ...
3
votes
2answers
222 views

Hadamard product and inertia

One of Schur's famous results says that if $A,B$ are positive semidefinite matrices, then the Hadamard (i.e. entrywise) product $A \circ B$ is also positive semidefinite. It's also true if "semi" is ...
1
vote
0answers
145 views

What is the Birkhoff norm of a Perron vector?

Let $A$ be a positive matrix. What is known about the Birkhoff norm of its Perron vector? By the Birkhoff norm of a vector $x$ I refer to the quantity $\frac{\max{x}}{\min{x}}$. P.S. This is ...
0
votes
0answers
150 views

vector equation

Suppose you have an equation of the form $Hx=Ky$, where $x,y$ are vectors of length $n,m$ respectively ($m>n$) and $H,K$ are matrices of orders $n \times n,n \times m$ respectively. Is there some ...
3
votes
3answers
731 views

upper bounds on a certain matrix norm

Is there some simple upper bound on $||(B^{-1}+A^{-1})^{-1}||$, where $A,B$ are $n \times n$ symmetric matrices?
1
vote
2answers
289 views

matrix stability criterion

I have a $5 \times 5$ parametric nonnegative matrix and want to show that it's stable (in the sense that all eigenvalues are positive). It is not symmetric, but I do know in advance that it has 5 real ...
1
vote
2answers
316 views

Sufficient conditions for inverse-positivity

I am trying to determine when a certain parametric matrix is inverse-positive (it's actually the one about which I asked in Explicit formula for Cholesky factorization in a special case, but the ...
2
votes
1answer
590 views

Explicit formula for Cholesky factorization in a special case

I have a positive definite matrix of the form $Q+sI-\alpha J$ ($s>2, 0 < \alpha <1$ and $J$ is the all-ones matrix), where $Q$ is "nice", nonnegative and known. I'd like to know if there is a ...
1
vote
0answers
174 views

matrix-theoretic terminology query

Is there an accepted term for the following property? Let $A$ be a real matrix such that all entries of the eigenvector corresponding to the least eigenvalue have the same sign. NOTES: (1) The case ...
2
votes
1answer
298 views

Null Space Perturbations

Hi, I face a problem some time now (not a homework problem) and I believe it is related to matrix perturbations and how the null space behaves in these cases. The distilled version of the ...