Summary
LetA andB be Hermitian matrices. The matrix-pair (A, B) is called “definite pair” and the corresponding eigenvalue problem,Ax=λBx is definite if\(c(A,B) \equiv \mathop {\min }\limits_{\left\| x \right\| = 1} \{ |x^H (A + iB)x|\} > 0\). The perturbation bounds for eigenspaces of a definite pair on every unitary-invariant matrix norm were obtained by imposing additional restrictions on the location of the generalized eigenvalues. Thus it gives a positive answer for an open question proposed by Stewart [7]. The famous Davis-Kahan sin θ theorems and sin 2θ Theorem [2] can also be deduced from the present results.
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Sun, Jg. The perturbation bounds for eigenspaces of a definite matrix-pair. Numer. Math. 41, 321–343 (1983). https://doi.org/10.1007/BF01418329
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DOI: https://doi.org/10.1007/BF01418329