Nearly defective complex eigenvalues in Bessel matrices

Eigensolutions of perturbed nearly defective matrices

Complex Eigenvalues 1. Complex Eigenvalues

In the previous note, we obtained the solutions to a homogeneous linear system with constant coefficients. x = A x under the assumption that the roots of its characteristic equation |A − λI| = 0, — i.e., the eigenvalues of A — were real and distinct. In this section we consider what to do if there are complex eigenval­ ues. Since the characteristic equation has real coefficients, its complex ro...

Eigenvalues and Condition Numbers of Complex Random Matrices

In this paper, the distributions of the largest and smallest eigenvalues of complex Wishart matrices and the condition number of complex Gaussian random matrices are derived. These distributions are represented by complex hypergeometric functions of matrix arguments, which can be expressed in terms of complex zonal polynomials. Several results are derived on complex hypergeometric functions and...

Eigenvalues and Pseudo-eigenvalues of Toeplitz Matrices

The eigenvalues of a nonhermitian Toeplitz matrix A are usually highly sensitive to perturbations, having condition numbers that increase exponentially with the dimension N. An equivalent statement is that the resolvent ( ZZ A)’ of a Toeplitz matrix may be much larger in norm than the eigenvalues alone would suggest-exponentially large as a function of N, even when z is far from the spectrum. B...

Nearly positive matrices

Nearly positive matrices are nonnegative matrices which, when premultiplied by orthogonal matrices as close to the identity as one wishes, become positive. In other words, all columns of a nearly positive matrix are mapped simultaneously to the interior of the nonnegative cone by mutiplication by a sequence of orthogonal matrices converging to the identity. In this paper, nearly positive matric...