Unifying unitary and hyperbolic transformations
Generalized unitary matrices are studied and described in known unified formulas with respect to arbitrary (in-)definite inner products. Here \(V\) is called \(S=\) diag\((\pm 1)\) unitary if \(V^*SV=S\) for the signature matrix \(S\). The unified transformations are applied to obtain generalized Cholesky factorizations of strongly nonsingular indefinite Hermitian matrices. Their condition numbers are studied with an eye on proper pivoting strategies. Multiplying by generalized Householder matrices is shown to be relationally stable. However, this paper does not mention any of the previous occurences and uses of generalized orthogonal matrices such as in algebraic group theory or in numerical matrix eigenvalue algorithms, as in the (unstable) HR algorithm [cf. \textit{A. Burse-Tertner}, ibid. 35, 155-173 (1981; Zbl 0462.65021)] or the DQR algorithm [cf. \textit{F. Uhlig}, Numer. Math. 76, No.~4, 515-553 (1997; Zbl 0883.65029)].
scientific article; zbMATH DE number 1533000