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*!

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). The matrix T is a trigonalization (or triangulation).

__Example:__ The Schur triangulation of the matrix $ M = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} $ gives $$ Q = \begin{bmatrix} −0.825 & 0.566 \\ 0.566 & −0.825 \end{bmatrix}, T = \begin{bmatrix} −0.372 & −1 \\ 0 & 5.372 \end{bmatrix} $$

There is always a decomposition of Schur, all matrices are trigonalizable, but not uniquely.

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}} $

The Schur decomposition makes it possible to simplify the form of the matrices and thus to facilitate the resolution of linear equations or any other problem using the matrix.

dCode retains ownership of the "Schur Decomposition (Matrix)" source code. Except explicit open source licence (indicated Creative Commons / free), the "Schur Decomposition (Matrix)" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Schur Decomposition (Matrix)" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Schur Decomposition (Matrix)" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!

Reminder : dCode is free to use.

The copy-paste of the page "Schur Decomposition (Matrix)" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!

Exporting results as a .csv or .txt file is free by clicking on the *export* icon

Cite as source (bibliography):

*Schur Decomposition (Matrix)* on dCode.fr [online website], retrieved on 2024-06-14,

schur,matrix,trigonalization,trigonalizable,triangulation,decomposition,unitary,triangular

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

© 2024 dCode — El 'kit de herramientas' definitivo para resolver todos los juegos/acertijos/geocaching/CTF.

Feedback