Characteristic polynomial: Difference between revisions
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In [[linear algebra]] the '''characteristic polynomial''' of a [[square matrix]] is a polynomial which has the [[eigenvalue]]s of the matrix as roots. | In [[linear algebra]] the '''characteristic polynomial''' of a [[square matrix]] is a polynomial which has the [[eigenvalue]]s of the matrix as roots. | ||
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where ''X'' is an indeterminate and ''I''<sub>''n''</sub> is an [[identity matrix]]. | where ''X'' is an indeterminate and ''I''<sub>''n''</sub> is an [[identity matrix]]. | ||
The characteristic polynomial is unchanged under [[similarity]], and hence be defined for an [[endomorphism]] of a [[vector space]], independent of choice of [[basis (linear algebra)|basis]]. | |||
==Properties== | ==Properties== | ||
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==Cayley-Hamilton theorem== | ==Cayley-Hamilton theorem== | ||
The '''Cayley-Hamilton theorem''' states that ''a matrix satisfies its own characteristic polynomial''. | The '''Cayley-Hamilton theorem''' states that ''a matrix satisfies its own characteristic polynomial''.[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 16:01, 26 July 2024
In linear algebra the characteristic polynomial of a square matrix is a polynomial which has the eigenvalues of the matrix as roots.
Let A be an n×n matrix. The characteristic polynomial of A is the determinant
where X is an indeterminate and In is an identity matrix.
The characteristic polynomial is unchanged under similarity, and hence be defined for an endomorphism of a vector space, independent of choice of basis.
Properties
- The characteristic polynomial is monic of degree n;
- The set of roots of the characteristic polynomial is equal to the set of eigenvalues of A.
Cayley-Hamilton theorem
The Cayley-Hamilton theorem states that a matrix satisfies its own characteristic polynomial.