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

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 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}} $

dCode retains ownership of the online 'Schur Decomposition (Matrix)' 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 or API access will be for free, same for Schur Decomposition (Matrix) download for offline use on PC, tablet, iPhone or Android !

Please, check our community Discord for help requests!

schur,matrix,triangularisation,triangularisable,triangulation,decomposition,unitary,triangular

Source : https://www.dcode.fr/matrix-schur-decomposition

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

Feedback

▲