Characteristic polynomial
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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.