Real eigenvalues of nonnegative matrices which commute with a symmetric matrix involution

Involution Matrices of Real Quaternions

An involution or anti-involution is a self-inverse linear mapping. In this paper, we will present two real quaternion matrices, one corresponding to a real quaternion involution and one corresponding to a real quaternion anti-involution. Moreover, properties and geometrical meanings of these matrices will be given as reflections in R^3.

Perturbing non-real eigenvalues of nonnegative real matrices

Extreme eigenvalues of real symmetric Toeplitz matrices

We exploit the even and odd spectrum of real symmetric Toeplitz matrices for the computation of their extreme eigenvalues, which are obtained as the solutions of spectral, or secular, equations. We also present a concise convergence analysis for a method to solve these spectral equations, along with an efficient stopping rule, an error analysis, and extensive numerical results.

On Reduced Rank Nonnegative Matrix Factorization for Symmetric Nonnegative Matrices

Let V ∈ R be a nonnegative matrix. The nonnegative matrix factorization (NNMF) problem consists of finding nonnegative matrix factors W ∈ R and H ∈ R such that V ≈ WH. Lee and Seung proposed two algorithms which find nonnegative W and H such that ‖V −WH‖F is minimized. After examining the case in which r = 1 about which a complete characterization of the solution is possible, we consider the ca...

Real symmetric matrices 1 Eigenvalues and eigenvectors

A matrix D is diagonal if all its off-diagonal entries are zero. If D is diagonal, then its eigenvalues are the diagonal entries, and the characteristic polynomial of D is fD(x) = ∏i=1(x−dii), where dii is the (i, i) diagonal entry of D. A matrix A is diagonalisable if there is an invertible matrix Q such that QAQ−1 is diagonal. Note that A and QAQ−1 always have the same eigenvalues and the sam...