Abstract
We compute the exact expected value of the squared condition number for the polynomial eigenvalue problem, when the input matrices have entries coming from the standard complex Gaussian distribution, showing that in general this problem is quite well conditioned.
Highlights
Recall the homogeneous generalized eigenvalue problem (GEVP): given two n \times n matrices A, B, find (\alpha, \beta ) \in \BbbC 2 \setminu \{ (0, 0)\} such that det(\beta A - \alpha B) = 0.The point (\alpha, \beta ) is called a generalized eigenvalue of (A, B) and the corresponding nonzero vectors x, y \in \BbbC n satisfying (\beta A - \alpha B)x = 0, y\ast (\beta A - \alpha B) = 0, are called the right and left eigenvectors of (A, B)
Since the hypotheses of Theorem 2.1 only involve the algebraic structure of the problem, we conclude that the expected square condition number for the standard Gaussian does not depend on the chosen Hermitian inner product on \scrI
As we will see, the same result on the expected value of the squared condition number obtained in section 3.1 holds if we endow \scrH d[X, Y ] with the Bombieri--Weyl Hermitian product which makes monomials of different degrees orthogonal and
Summary
Since the hypotheses of Theorem 2.1 only involve the algebraic structure of the problem, we conclude that the expected square condition number for the standard Gaussian does not depend on the chosen Hermitian inner product on \scrI (see Examples 3.1 and 3.2). As we will see, the same result on the expected value of the squared condition number obtained in section 3.1 holds if we endow \scrH d[X, Y ] with the Bombieri--Weyl Hermitian product which makes monomials of different degrees orthogonal and
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