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Schur decomposition
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== Generalized Schur decomposition == Given square matrices ''A'' and ''B'', the '''generalized Schur decomposition''' factorizes both matrices as <math>A = QSZ^*</math> and <math>B = QTZ^*</math>, where ''Q'' and ''Z'' are [[unitary matrix|unitary]], and ''S'' and ''T'' are [[upper triangular]]. The generalized Schur decomposition is also sometimes called the '''QZ decomposition'''.<ref name=Golub1996/>{{rp|p=375}} <ref>Daniel Kressner: "Numerical Methods for General and Structured Eigenvalue Problems", Chap-2, Springer, LNCSE-46 (2005).</ref> The generalized [[eigenvalue]]s <math>\lambda</math> that solve the [[Eigendecomposition of a matrix#Additional topics|generalized eigenvalue problem]] <math>A\mathbf{x}=\lambda B\mathbf{x}</math> (where '''x''' is an unknown nonzero vector) can be calculated as the ratio of the diagonal elements of ''S'' to those of ''T''. That is, using subscripts to denote matrix elements, the ''i''th generalized eigenvalue <math>\lambda_i</math> satisfies <math>\lambda_i = S_{ii} / T_{ii}</math>.
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