Characteristic polynomial
Polynomial whose roots are the eigenvalues of a matrix
Summary
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any basis. The characteristic equation, also known as the determinantal equation, is the equation obtained by equating the characteristic polynomial to zero.
Originally created by AxelBoldt
4/29/2003, 3:06:50 AM
Modified
5/31/2026, 6:02:02 PM
Recent revisions
Added the Linear algebra template.
/* Characteristic polynomial of a product of two matrices */ proof environment
[[Wikipedia:OABOT|Open access bot]]: arxiv updated in citation with #oabot.
Added information of theoretical complexity
Split equation to multiple lines
two links
rewrote so two different links are next to each other
Reverted 1 edit by [[Special:Contributions/Goodphy|Goodphy]] ([[User talk:Goodphy|talk]])
/* Formal definition */ Fixed a grammatical error.
/* Characteristic polynomial of a product of two matrices */ ce
/* Characteristic polynomial of a product of two matrices */ Oops, had deleted too much.
Undid revision [[Special:Diff/1304580550|1304580550]] by [[Special:Contributions/Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]])
/* Characteristic polynomial of a product of two matrices */ rewrote wrong stuff
/* Characteristic polynomial of a product of two matrices */ Added that AB and BA are similar if B is non-singular (invertible).
Added "more efficiently" and also a request for clarification, based on your own edit?
/* Properties */ Improved a sentence regarding the Faddeev–LeVerrier algorithm.
/* Formal definition */ Mentioned how a different characteristic polynomial definition is monic only if the size of the corresponding matrix is even number-by-even number.
We do not need to reiterate this over and over again
/* Properties */ Made a word linked to the corresponding section of a page.
/* Properties */
