Tool to calculate eigenvalues of a matrix. The eigenvalues of a matrix are values that allow to reduce the associated endomorphisms.

Eigenvalues of a 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 eigenvalues of a matrix. The eigenvalues of a matrix are values that allow to reduce the associated endomorphisms.

**Eigenvalues** are numbers that characterize a matrix. These numbers are important because, associated with their eigenvectors, they make it possible to express the matrix in a simpler form, which facilitates the calculations.

for any square matrix $ M $ of size $ m \times m $, **eigenvalues** are generally called lambda $ \lambda $ and associated with an eigenvector $ v $ if $$ M.v = \lambda v \iff (M-\lambda I_m).v = 0 $$ with $ I_m $ the identity matrix (of size $ m $).

Practically, the **eigenvalues** of $ M $ are the roots of its characteristic polynomial $ P $.

An **eigenvalue** of a matrix is always associated with an eigenvector. Use the eigenvectors calculator proposed by dCode.

To find the **eigenvalues** of a matrix, calculate the roots of its characteristic polynomial.

Example: The 2x2 matrix $ M=\begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix} $ has for characteristic polynomial $ P(M) = x^2 − 4x − 5 = (x+1)(x-5) $. The roots of $ P $ are found by the calculation $ P(M)=0 \iff x= -1 \mbox{ or } x = 5 $. The **eigenvalues** of the matrix $ M $ are $ -1 $ and $ 5 $.

NB : The eigenvector associated are $ \begin{bmatrix} 1 \\ 2 \end{bmatrix} $ for $ 5 $ and $ \begin{bmatrix} -1 \\ 1 \end{bmatrix} $ for $ -1 $

**Eigenvalues** are called eigen because it is as German word which means proper, characteristic.

dCode retains ownership of the source code of the script Eigenvalues of a Matrix online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be released for free. To download the online Eigenvalues of a Matrix script for offline use on PC, iPhone or Android, ask for price quote on contact page !

eigenvalue,matrix,eigenvector,space,characteristic

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

© 2019 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaches. dCode

Feedback

▲