Tool to calculate Schur decomposition (or Schur triangulation) that makes it possible to write any numerical square matrix into a multiplication of a unitary matrix and an upper triangular matrix.

Schur Decomposition (Matrix) - dCode

Tag(s) : Matrix

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Tool to calculate Schur decomposition (or Schur triangulation) that makes it possible to write any numerical square matrix into a multiplication of a unitary matrix and an upper triangular 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}} $

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