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Characteristic polynomial

This online calculator calculates coefficients of characteristic polynomial of a square matrix using Faddeev–LeVerrier algorithm

Timur2019-06-17 05:34:04

In linear algebra, the characteristic polynomial of an n×n square matrix A is a polynomial that is invariant under matrix similarity and has the eigenvalues as roots. The polynomial pA(λ) is monic (its leading coefficient is 1), and its degree is n. The calculator below computes coefficients of a characteristic polynomial of a square matrix using the Faddeev–LeVerrier algorithm. You can find theory and formulas below the calculator.

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Characteristic polynomial

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Characteristic polynomial
 
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Characteristic polynomial

Given a square matrix A, we want to find a polynomial whose zeros are the eigenvalues of A. For a diagonal matrix A, the characteristic polynomial is easy to define: if the diagonal entries are a1, a2, a3, etc., then the characteristic polynomial will be:

(t-a_1)(t-a_2)(t-a_3)\cdots,

This works because the diagonal entries are also the eigenvalues of this matrix.

For a general matrix A, one can proceed as follows. A scalar λ is an eigenvalue of A if and only if there is an eigenvector v ≠ 0 such that

A \mathbf{v} = \lambda \mathbf{v},

or

(\lambda I - A)\mathbf{v} = 0\,

(where I is the identity matrix).

Since v is non-zero, the matrix λ I − A is singular (non-invertible), which means that its determinant is 0. Thus the roots of the function det(λ I − A) are the eigenvalues of A, and it is clear that this determinant is a polynomial in λ.1

In matrix form polynomial in λ looks like this:

p(\lambda)=det \begin{bmatrix}\lambda - a_{11}&a_{12}&\dots &a_{1n}\\a_{21}&\lambda - a_{22}&\dots &a_{2n}\\ \dots & \dots & \dots & \dots \\a_{n1}&a_{n2}&\dots &\lambda - a_{nn}\end{bmatrix}

In scalar form

p(\lambda )\equiv \det(\lambda I_{n}-A)=\sum _{k=0}^{n}c_{k}\lambda ^{k}~,

where, cn = 1 and c0 = (−1)n det A.

The coefficients can be found using the recursive Faddeev–LeVerrier algorithm (first published in 1840 by Urbain Le Verrier, in present form redeveloped by Dmitry Konstantinovich Faddeev and others).

Faddeev–LeVerrier algorithm

The coefficients of the characteristic polynomial are determined recursively from the top down, by dint of the auxiliary matrices M2,

{\begin{aligned}M_{0}&\equiv 0&c_{n}&=1\qquad &(k=0)\\M_{k}&\equiv AM_{k-1}+c_{n-k+1}I\qquad \qquad &c_{n-k}&=-{\frac {1}{k}}\mathrm {tr} (AM_{k})\qquad &k=1,\ldots ,n~.\end{aligned}}

Thus,

{\displaystyle M_{1}=I~, \quad c_{n-1}=-\mathrm {tr} A=-c_{n}\mathrm {tr} A;} \\{\displaystyle M_{2}=A-I\mathrm {tr} A, \\ \quad c_{n-2}=-{\frac {1}{2}}{\Bigl (}\mathrm {tr} A^{2}-(\mathrm {tr} A)^{2}{\Bigr )}=-{\frac {1}{2}}(c_{n}\mathrm {tr} A^{2}+c_{n-1}\mathrm {tr} A);} \\{\displaystyle M_{3}=A^{2}-A\mathrm {tr} A-{\frac {1}{2}}{\Bigl (}\mathrm {tr} A^{2}-(\mathrm {tr} A)^{2}{\Bigr )}I,} \\{\displaystyle c_{n-3}=-{\tfrac {1}{6}}{\Bigl (}(\operatorname {tr} A)^{3}-3\operatorname {tr} (A^{2})(\operatorname {tr} A)+2\operatorname {tr} (A^{3}){\Bigr )}=-{\frac {1}{3}}(c_{n}\mathrm {tr} A^{3}+c_{n-1}\mathrm {tr} A^{2}+c_{n-2}\mathrm {tr} A);}

etc.,

{\displaystyle M_{m}=\sum _{k=1}^{m}c_{n-m+k}A^{k-1}~,} \\{\displaystyle c_{n-m}=-{\frac {1}{m}}(c_{n}\mathrm {tr} A^{m}+c_{n-1}\mathrm {tr} A^{m-1}+...+c_{n-m+1}\mathrm {tr} A)=-{\frac {1}{m}}\sum _{k=1}^{m}c_{n-m+k}\mathrm {tr} A^{k}~;...}

The calculator uses this algorithm to compute the coefficients. It can also output auxiliary matrix M for each step.

  1. Wikipedia - Characteristic polynomial ↩

  2. Wikipedia - Faddeev–LeVerrier algorithm ↩

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