Tool to calculate a matrix triangularization / trigonalization in order to write a square matrix in a composition of a superior triangular matrix and a unitary matrix.

Matrix Trigonalization - dCode

Tag(s) : Matrix

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Tool to calculate a matrix triangularization / trigonalization in order to write a square matrix in a composition of a superior triangular matrix and a unitary matrix.

**Matrix Trigonalisation** of a square matrix $ M $ consists of writing the matrix in the form: $$ M = Q.T.Q ^ {- 1} $$

with $ T $ a ** upper triangular matrix ** and $ Q $ a unitary matrix (i.e. $ Q ^ *. Q = I $ identity matrix).

This calculation, also called Schur decomposition, uses the eigenvalues of the matrix as values of the diagonal.

Schur's theorem indicates that there is always at least one decomposition on $ \mathbb{C} $ (so the matrix is trigonalizable/triangularizable).

This trigonalization only applies to numerical or complex square matrices (without variables).

dCode uses Schur decomposition via computer algorithms such as QR decomposition.

Manually, for a matrix matrix $ M $, calculate its eigenvalues $ \ Lambda_i $ and deduce an eigenvector $ u_1 $

Calculate its normalized value in an orthonormal base $ {u_1, v_2} $ in order to obtain $ U = [u_1, v_2] $

Then express the matrix in the orthonormal base $ A_{{u_1,v_2}} = U^{-1}.A.U = U^{T}.A.U $

Finally, repeat the operation for each eigenvector in order to obtain the triangular matrix.

For a 2x2 matrix, only one operation is necessary and $ T = A_{{u_1,v_2}} $

__Example:__ Schur triangulation for the matrix $ M = \begin{pmatrix} 4 & 3 \\ 2 & 1 \end{pmatrix} $ gives $$ Q = \begin{pmatrix} 0.909 & 0.415 \\ -0.415 & 0.909 \end{pmatrix}, T = \begin{pmatrix} 5.37 & −1 \\ 0 & −0.37 \end{pmatrix} $$

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