Editing Matrix decomposition (section)

=== QZ decomposition ===
{{main|QZ decomposition}}
*Also called: ''generalized Schur decomposition''
*Applicable to: [[square matrix|square matrices]] ''A'' and ''B''
*Comment: there are two versions of this decomposition: complex and real.
*Decomposition (complex version): <math>A=QSZ^*</math> and <math>B=QTZ^*</math> where ''Q'' and ''Z'' are [[unitary matrix|unitary matrices]], the * superscript represents [[conjugate transpose]], and ''S'' and ''T'' are [[upper triangular]] matrices.
*Comment: in the complex QZ decomposition, the ratios of the diagonal elements of ''S'' to the corresponding diagonal elements of ''T'', <math>\lambda_i = S_{ii}/T_{ii}</math>, are the generalized [[eigenvalue]]s that solve the [[Eigendecomposition of a matrix#Additional topics|generalized eigenvalue problem]] <math>A \mathbf{v} = \lambda B \mathbf{v}</math> (where <math>\lambda</math> is an unknown scalar and '''v''' is an unknown nonzero vector).
*Decomposition (real version): <math>A=QSZ^\mathsf{T}</math> and <math>B=QTZ^\mathsf{T}</math> where ''A'', ''B'', ''Q'', ''Z'', ''S'', and ''T'' are matrices containing real numbers only. In this case ''Q'' and ''Z'' are [[orthogonal matrix|orthogonal matrices]], the ''T'' superscript represents [[matrix transpose|transposition]], and ''S'' and ''T'' are [[block matrix|block upper triangular]] matrices. The blocks on the diagonal of ''S'' and ''T'' are of size 1×1 or 2×2.