MATRIX POLYNOMIALS, SIMILAR OPERATORS, AND THE IMAGINARY AXIS EIGENVALUES OF A MATRIX DELAY EQUATION

We present a new approach to determining the imaginary axis eigenvalues of a matrix delay equation. With a full rank delay coefficient matrix, the approach requires computation of the generalized eigenvalues of a pair of matrices which are a quarter of the size used in currently known matrix-based or operator approaches. These matrices have square of n for both row and column length, improved from twice those lengths, for an equation having matrix n X n coefficients with rank n delay matrix. Considering problem dimension and computational demands, this brings matrix-based approaches more in synchrony with the well-known scalar substitution approach. Given the symmetries evident in our approach, problems previously done with computation alone can now be considered with insight, and sometimes completed with simple calculations. Some of the computational and mathematical features of the approach are displayed in a section of examples.