One of the most efficient methods for solving the polynomial eigenvalue problem (PEP) is the Sakurai-Sugiura method with Rayleigh-Ritz projection (SS-RR), which finds the eigenvalues contained in a certain domain using the contour integral. The SS-RR method converts the original PEP to a small projected PEP using the Rayleigh-Ritz projection. However, the SS-RR method suffers from backward instability when the norms of the coefficient matrices of the projected PEP vary widely. To improve the backward...

In this paper, the concepts of indecomposable matrices and fully indecomposable matrices over a distributive lattice $L$ are introduced, and some algebraic properties of them are obtained. Also, some characterizations of the set $F_n\left(L\right)$ of all $n\times n$ fully indecomposable matrices as a subsemigroup of the semigroup $H_n\left(L\right)$ of all $n\times n$ Hall matrices over the lattice $L$ are given.

Let $a=(a_1,a_2,...,a_n)$ be a nonincreasing sequence of positive real numbers. Denote by $S=\{1,2,...,n\}$ the index set and by $J_k=\{I=\{r_1,r_2,...,r_k\}$, $1\le r_1<r_2<\cdots <r_k\le n\}$ the set of all subsets of $S$ of cardinality $k$, $1\le k\le n-1$. In addition, denote by $a_I=a_{r_1}+a_{r_2}+\cdots +a_{r_k}$, $1\le k\le n-1$, $1\le r_1<r_2<\cdots <r_k\le n$, the sum of $k$ arbitrary elements of sequence $a$, where $a_{I_1}=a_1+a_2+\cdots +a_k$ and $a_{I_n}=a_{n-k+1}+a_{n-k+2}+\cdots +a_n$. We consider bounds of the quantities $R{S}_k\left(a\right)=a_{I_1}/a_{I_n}$, $L{S}_k\left(a\right)=a_{I_1}-a_{I_n}$ and $S_{k,\alpha }\left(a\right)={\sum }_{I\in J_k}{a}_I^{\alpha }$ in terms of $A={\sum }_{i=1}^n{a}_i$ and $B={\sum }_{i=1}^n{a}_i^2$. Then we use the obtained results to generalize some results regarding Laplacian and normalized Laplacian eigenvalues of graphs.

Let S be a given set consisting of some Hermitian matrices with the same size. We say that a matrix A ∈ S is maximal if A − W is positive semidefinite for every matrix W ∈ S. In this paper, we consider the maximal and minimal inertias and ranks of the Hermitian matrix function f(X,Y) = P − QXQ* − TYT*, where * means the conjugate and transpose of a matrix, P = P*, Q, T are known matrices and for X and Y Hermitian solutions to the consistent matrix equations AX =B and YC = D respectively. As applications,...

It has been known for a long time that for birth-and-death processes started in zero the first passage time of a given level is distributed as a sum of independent exponentially distributed random variables, the parameters of which are the negatives of the eigenvalues of the stopped process. Recently, Diaconis and Miclo have given a probabilistic proof of this fact by constructing a coupling between a general birth-and-death process and a process whose birth rates are the negatives of the eigenvalues,...

We collect certain useful lemmas concerning the characteristic map, ${𝒢L}_n$-invariant sets of matrices, and the relative codimension. We provide a characterization of rank varieties in terms of the characteristic map as well as some necessary and some sufficient conditions for linear subspaces to allow the dominant restriction of the characteristic map.