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Matrix Trigonalization

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.

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Matrix Trigonalization -

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# Matrix Trigonalization

## Matrix Trigonalization Calculator

### What is the Matrix Trigonalization? (Definition)

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).

### How to calculate the triangular matrix?

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 $$Q = \begin{bmatrix} 0.909 & 0.415 \\ -0.415 & 0.909 \end{bmatrix}, T = \begin{bmatrix} 5.37 & −1 \\ 0 & −0.37 \end{bmatrix}$$

### How to demonstrate that a matrix is trigonalizable?

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.

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Matrix Trigonalization on dCode.fr [online website], retrieved on 2023-09-23, https://www.dcode.fr/matrix-trigonalization

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