Tool to diagonalize a matrix. The diagonalization of a matrix consists of writing it in a base where its elements outside the diagonal are null.
Matrix Diagonalization - 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!
A diagonal matrix is a matrix whose elements out of the trace (the main diagonal) are all null (zeros).
Example: $$ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix} $$
Diagonalization is a transform used in linear algebra so that it allows performing easier calculations.
A matrix is diagonalizable if there exists an invertible matrix $ P $ and a diagonal matrix $ D $ such that $ M = PDP^{-1} $
To diagonalize a matrix, a diagonalisation method consists in calculating its eigenvectors and its eigenvalues.
Example: The matrix $$ M = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix} $$ has for eigenvalues $ 3 $ and $ -1 $ and eigenvectors respectively $ \begin{bmatrix} 1 \\ 1 \end{bmatrix} $ and $ \begin{bmatrix} -1 \\ 1 \end{bmatrix} $
The diagonal matrix $ D $ is composed of eigenvalues.
Example: $$ D = \begin{bmatrix} 3 & 0 \\ 0 & -1 \end{bmatrix} $$
The invertible matrix $ P $ is composed of the eigenvectors respectively in the same order of the columns than its associated eigenvalues.
P must be a normalized matrix.
Example: $$ P = \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix} $$ Normalization of P: $$ P = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix} $$
A matrix is not diagonalizable if it does not have as many dimensions as distinct eigenvectors.
Example: The matrix of dimension 2: $$ M = \begin{bmatrix} 5 & 1 \\ 0 & 5 \end{bmatrix} $$ has a double eigenvalue: $ 5 $ and therefore a single eigenvector $ \begin{bmatrix} 1 \\ 0 \end{bmatrix} $ so it is not diagonalizable.
Example: A 3x3 matrix with a triple eigenvalue therefore a single eigenvector is not diagonalizable.
Calculate the inverse of the matrix $ P $
Diagonalization should give $ PDP^{-1} = M $
dCode retains ownership of the "Matrix Diagonalization" source code. Except explicit open source licence (indicated Creative Commons / free), the "Matrix Diagonalization" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Matrix Diagonalization" 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 Diagonalization" 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 "Matrix Diagonalization" or any of its results, is allowed (even for commercial purposes) as long as you cite dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Matrix Diagonalization on dCode.fr [online website], retrieved on 2023-10-01,