Schur Decomposition (Matrix)

The Schur decomposition of a square matrix $ M $ is its writing in the following form (also called Schur form): $$ M = Q.T.Q^{-1} $$

with $ Q $ a unitary matrix (such as $ Q^*.Q=I $) and $ T $ is an upper triangular matrix whose diagonal values are the eigenvalues of the matrix.

This decomposition only applies to numerical square matrices (no variables)

Example: The Schur triangulation of the matrix $ M = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix} $ gives $$ Q = \begin{pmatrix} −0.825 & 0.566 \\ 0.566 & −0.825 \end{pmatrix}, T = \begin{pmatrix} −0.372 & −1 \\ 0 & 5.372 \end{pmatrix} $$

There is always a decomposition of Schur, but it is not necessarily unique.

dCode uses computer algorithms involving QR decomposition.

Manually, find a proper vector $ u_1 $ of the matrix $ M $ by calculating its eigenvalues $ \Lambda_i $. Calculate its normalized value and an orthonormal basis $ {u_1, v_2} $ to obtain $ U = [u_1, v_2] $. Express the matrix $ M $ in the orthonormal basis $ A_{{u_1, v_2}} = U^{-1}.A.U = U^{T}.A.U $. Repeat the operation for each eigenvector to obtain the triangular matrix. NB: for a 2x2 matrix, only one operation is necessary and $ T = A_{{u_1, v_2}} $