Characteristic polynomial: Difference between revisions
Jump to navigation
Jump to search
imported>Richard Pinch (New entry, just a stub, with anchors) |
imported>Chris Day No edit summary |
||
| (3 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
{{subpages}} | |||
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. | ||
| Line 6: | Line 7: | ||
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== | ||
Latest revision as of 22:35, 17 February 2009
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.