Matrix Trigonalization

Question 1

What is the Matrix Trigonalization? (Definition)

Answer

Matrix Trigonalisation (sometimes names triangularization) of a square matrix $ M $ consists of writing the matrix in the form: $$ M = Q.T.Q^{-1} $$

with $ T $ an 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).

Question 2

How to calculate the triangular matrix?

Answer

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 this operation for each of the eigenvectors 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 triangularisation for the matrix $ M = \begin{bmatrix} 4 & 3 \\ 2 & 1 \end{bmatrix} $ gives $$ Q = \begin{bmatrix} 0.909 & 0.415 \\ -0.415 & 0.909 \end{bmatrix}, T = \begin{bmatrix} 5.37 & −1 \\ 0 & −0.37 \end{bmatrix} $$

Question 3

How to demonstrate that a matrix is trigonalizable?

Answer

Calculate the characteristic polynomial $ P $ of the matrix. It will be trigonalisable if and only if $ P $ is split. That is, it is representable as a product of first degree polynomials.

Matrix Trigonalization