Search for a tool
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.

Results

Matrix Trigonalization -

Tag(s) : Matrix

Share
Share
dCode and more

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!


Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!


Feedback and suggestions are welcome so that dCode offers the best 'Matrix Trigonalization' tool for free! Thank you!

Matrix Trigonalization

Matrix Trigonalization Calculator

Loading...
(if this message do not disappear, try to refresh this page)

Answers to Questions (FAQ)

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.

Source code

dCode retains ownership of the "Matrix Trigonalization" source code. Except explicit open source licence (indicated Creative Commons / free), the "Matrix Trigonalization" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Matrix Trigonalization" 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 "Matrix Trigonalization" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.

Cite dCode

The copy-paste of the page "Matrix Trigonalization" 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):
Matrix Trigonalization on dCode.fr [online website], retrieved on 2024-06-16, https://www.dcode.fr/matrix-trigonalization

Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Questions / Comments

Feedback and suggestions are welcome so that dCode offers the best 'Matrix Trigonalization' tool for free! Thank you!


https://www.dcode.fr/matrix-trigonalization
© 2024 dCode — El 'kit de herramientas' definitivo para resolver todos los juegos/acertijos/geocaching/CTF.
 
Feedback