Fast Generation of Real-Symmetric Matrices and their Exact Eigenpairs

Abstract

We propose a method to rapidly generate matrices for real-symmetric eigenproblems. The proposed method produces a reproducible matrix with explicit eigenpairs, where the distribution of the eigenvalues can be controlled by a user. All elements of the generated matrix are rigorous floating-point numbers and can be represented in simple expressions involving the exact eigenvalues. The exact eigenpairs of the generated matrix are known in advance; thus, the proposed method contributes to the validation of errors in approximate eigenpairs. Several constraints on the matrix generated by the proposed method, were produced theoretically.