Tagged Questions

0answers
28 views

Gershgorin interval of an eigenvalue and the largest coordinate of the corresponding eigenvector

Let $A=(a_{ij})$ be a $n\times n$ -- symmetric matrix with positive diagonal entries. The smallest eigenvalue, $\lambda_1$, is simple, and the corresponding unit eigenvector has …
2answers
198 views

spectral radius monotonicity

I encountered an inequality when reading a paper. Can someone help to show how to prove it? Let be the spectral radius of matrix $A$ or \$\rho(A)=\max{|\lambda|, \lambda \text{ are …
2answers
167 views

Dimension of incomplete matrix over finite fields.

Hi, Suppose one has an incompletely specified $2^n \times 2^n$ matrix over some fixed finite field $\mathbb{F}_{p^k}$. In fact, one knows that the diagonal entries are zero and al …
1answer
45 views

On solution of a class of discrete-time  Lyapunov equation

Hello members, let's consider the following equation $$X=F_{1}XF_{1}^{T}+...+F_{p}XF_{p}^{T}+C$$ where $p$ is an positive integer and $C$ is a known positive semidefinite matrix. I …
2answers
224 views

tracial triples

Say that a triple of real numbers $(a,b,c)$ is a realizable triple if there are matrices $A,B\in SL_2(\mathbb{R})$ such that $tr (A)=a$, $tr (B)=b$, and $tr (AB)=c$. Question: what …
1answer
71 views

On solution of a discrete-time equation

Hello, members. I have a problem for the following problem when I derive an optimization algorithm for stochastic singular systems S(k+1)=A(k)S(k)A^{T}(k)+R(k)+F(k)S(k+1)F^{T}( …
1answer
70 views

An Interesting variant of Rayleigh Quotient

Let $A$ and $B$ be two given hermitian positive semi-definite matrices, then what is the solution for \begin{align} \max_{x\neq 0}\frac{x^HAx}{x^HBx+1}. \end{align} I am looking f …
1answer
74 views

On solution of a recursion with rectangular matrices

Greetings to members here. The question is how to calculate the solution $S(k)$ of the following recursive equation $$J(k)S(k+1)J^{T}(k)=A(k)S(k)A^{T}(k)+R(k)$$ where $J$ and $A$ …
1answer
72 views

Schur product, partial order

Let $A, B$ be positive definite matrices. Then $A^r\circ B^r \le (A\circ B)^r$ for $0\le r\le 1$, where $\circ$ is Schur product. Here the inequality is in the sense of Loewner par …
1answer
33 views

Feasibility of a given set of homogenuous nonconvex quadratic inequality constraints

Let $C_1$,$C_2$,...$C_N$ be $M \times M$ indefinite hermitian matrices. What can we say about the following quadratic constriants \begin{align} w^{H}C_1w>0 \\ w^{H}C_2w>0 \\ ...~~~ …
2answers
191 views

Efficient computation of Markov chain transition probability matrix

Consider a continuous Markov chain $X = (X_t)$ on a finite state space and let $Q$ be the (given) transition rate matrix. This matrix is very sparse, with non-zero values on 3 diag …
0answers
87 views

Examples of functions from matrices to real numbers with certain properties

Let $M(\mathbb{R})$ be the set of all matrices (of any size) over $\mathbb{R}$. Let $v : M(\mathbb{R}) \rightarrow \mathbb{R}$ be a function which satisfies the following 5 proper …
2answers
73 views

Strictly positive definite autocovariance function of fGn

Hi, let $\gamma(k) = 1/2 (|k+1|^{2H} + |k-1|^{2H}-2|k|^{2H}),k\in\mathbb{Z},$ be autocovariance function of fractional Gaussian noise where $H\in(0,1)$ is parameter. I want to sh …
0answers
101 views

Products of matrices of a certain form

Are $n \times n$ matrices of the form $$\pmatrix{1&1&1&1 \cr x&1&1&1 \cr x&x&1&1 \cr x&x&x&1}$$ studied anywhere? I am interested in …
0answers
238 views

Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials

While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so afte …

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