Tool to calculate the characteristic polynomial of a matrix. The characteristic polynomial of a matrix M is computed as the determinant of (X.I-M).

Characteristic Polynomial of a Matrix - dCode

Tag(s) : Mathematics,Algebra,Symbolic Computation

dCode is free and its tools are a valuable help in games, puzzles and problems to solve every day!

You have a problem, an idea for a project, a specific need and dCode can not (yet) help you? You need custom development? *Contact-me*!

This page is using the new English version of dCode, *please make comments* !

Sponsored ads

Tool to calculate the characteristic polynomial of a matrix. The characteristic polynomial of a matrix M is computed as the determinant of (X.I-M).

Consider \( M \) a square matrix of size \( n \), the characteristic polynomial \( P \) of the matrix \( M \) is the polynomial defined by $$ P(M) = \det( X.I_n - M ) $$ with \( I_n \) the identity matrix of size \( n \).

$$ M=\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \Rightarrow X.I_n - M = \begin{pmatrix} x-1 & -2 \\ -3 & x-4 \end{pmatrix} \Rightarrow \det(X.I_n - M) = (x-1)(x-4)-((-2)\times(-3)) = x^2-5x-2 $$

If \( M \) is a diagonal matrix with \( \lambda_1, \lambda_2, \ldots, \lambda_n \) as diagonal elements, then $$ P(M) = \det( X.I_n - M ) = (X-\lambda_1)(X-\lambda_2)\ldots(X-\lambda _n) $$

The calculation of the characteristic polynomial of a square matrix of order 2 can be written using the trace of the matrix M : $$ P(M) = \det( X.I_2 - M ) = X^2 - \operatorname{Tr}(M)X+ \det(M) $$

For a 3x3 matrix M the characteristic polynomial may be written using the trace of the adjugate matrix M: $$ P(M) = \det( X.I_2 - M ) = X^3 - \rm{Tr}(M)X^2 + \rm{Tr}( \rm{com}(M))X - \det(M) $$

dCode retains ownership of the source code of the script Characteristic Polynomial of a Matrix. Except explicit open source licence (free / freeware), any algorithm, applet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any snippet or function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in PHP (or Java, C#, Python, Javascript, etc.) which dCode owns rights can be transferred after sales quote. So if you need to download the Characteristic Polynomial of a Matrix script for offline use, for you, your company or association, see you on contact page !

characteristic,polynomial,matrix,eigenvalue,eigenvector,determinant

Source : http://www.dcode.fr/matrix-characteristic-polynomial

© 2017 dCode — The ultimate 'toolkit' website to solve every problem. dCode