Characteristic and minimal polynomials
The characteristic polynomial of $A$ is $p(\lambda) = \det(A - \lambda I)$. Its roots are the eigenvalues. The minimal polynomial $m(\lambda)$ is the monic polynomial of least degree such that $m(A) = 0$. The minimal polynomial divides the characteristic polynomial and has the same roots (but possibly with lower multiplicities).
Parents
Children
- Definition of applying a polynomial to a square matrix
- Definition of characteristic polynomial of a matrix
- Definition of characteristic equation of a matrix
- Definition of minimal polynomial of a matrix
- The minimal polynomial of a square matrix exists and is unique.
- The eigenvalues of a matrix are the roots/solutions of its characteristic polynomial/equation.
- The characteristic polynomial applied to the matrix gives the 0 matrix.
- The Cayley-Hamilton theorem for a matrix.