Matrix Minimal Polynomial

The matrix minimal polynomial of a matrix is the monic polynomial in of smallest degree such that

(1)

The minimal polynomial divides any polynomial with and, in particular, it divides the characteristic polynomial.

If the characteristic polynomial factors as

(2)

then its minimal polynomial is given by

(3)

for some positive integers , where the satisfy .

For example, the characteristic polynomial of the zero matrix is , while its minimal polynomial is . However, the characteristic polynomial and minimal polynomial of

(4)

are both .

The minimal polynomial of a square matrix in the variable is implemented in the Wolfram Language as MatrixMinimalPolynomial[a, x].