On worst-case condition numbers of a nondefective multiple eigenvalue (Q1347052)

This paper shows that the quantities \(c_ j (A; \lambda_ 1) \equiv (\prod^ j_{k=1} c_ k (A; \lambda_ 1))^{1/j}\), \(j = 1, \dots, r\), are the worst-case condition numbers of the multiple eigenvalue \(\lambda_ 1\), where \(\lambda_ 1\) is a nondefective multiple eigenvalue of multiplicity \(r\) of an \(n \times n\) complex matrix \(A\), and \(c_ 1 (A; \lambda_ 1) \geq \cdots \geq c_ r (A; \lambda_ 1)\) are the secants of the canonical angles between the left and right invariant subspaces of \(A\) corresponding to the multiple eigenvalue \(\lambda_ 1\). A simple example to test its conclusions is illustrated in the last section.