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

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!

A suggestion ? a feedback ? a bug ? an idea ? *Write to dCode*!

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** (sometimes names triangularization) 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 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{pmatrix} 4 & 3 \\ 2 & 1 \end{pmatrix} $ gives $$ Q = \begin{pmatrix} 0.909 & 0.415 \\ -0.415 & 0.909 \end{pmatrix}, T = \begin{pmatrix} 5.37 & −1 \\ 0 & −0.37 \end{pmatrix} $$

dCode retains ownership of the online 'Matrix Trigonalization' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) no data, script, copy-paste, or API access will be for free, same for Matrix Trigonalization download for offline use on PC, tablet, iPhone or Android !

Please, check our community Discord for help requests!

triangularization,triangularizable,triangularisation,triangular,schur,matrix

Source : https://www.dcode.fr/matrix-trigonalization

© 2021 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.

Feedback

▲