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I have used a matrix decomposition in my work that a different author calls “asymmetric Schur decomposition”. Now I am trying to find a reference for this and found that the Schur decomposition is usually of a different form, namely with three matrices. From the overall form it more looks like an LU-decomposition, but Mathematica gives me a different result when decomposing this matrix.

Could you tell me what type of decomposition this is?

$$ M = \begin{pmatrix} M_\mathrm{ee} & M_\mathrm{eo} \\ M_\mathrm{oe} & M_\mathrm{oo} \end{pmatrix} = \begin{pmatrix} M_\mathrm{ee} & \mathbf 0 \\ M_\mathrm{oe} & \mathbf 1 \end{pmatrix} \begin{pmatrix} \mathbf 1 & M_\mathrm{ee}^{-1} M_\mathrm{eo} \\ \mathbf 0 & M_\mathrm{oo} - M_\mathrm{oe} M_\mathrm{ee}^{-1} M_\mathrm{eo} \end{pmatrix} $$

Martin Ueding
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It is just an LU decomposition in which the diagonal of L was not turned into $1$s. LU is not really unique if only the triangular shapes are required.

Think of an LU decomposition as performing row reduction of a matrix to upper triangular form. It can be done in many ways.

Hellen
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