Characteristic polynomial: Difference between revisions
Jump to navigation
Jump to search
imported>Chris Day No edit summary |
mNo edit summary |
||
Line 15: | Line 15: | ||
==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 17:01, 26 July 2024
![](http://s9.addthis.com/button1-share.gif)
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.