`minpoly`

Minimal polynomial of a matrix.

Syntax

p = minpoly(A)

Coefficient vector of the minimal polynomial of A, highest power first.

p = minpoly(A, x)

Minimal polynomial as an expression in the symbolic variable x.

Description

The minimal polynomial of a matrix A is the monic polynomial of lowest degree that vanishes at A. It divides the characteristic polynomial and has the same roots, but each root may appear with a smaller multiplicity. The two coincide when the matrix has no repeated eigenvalue with several independent eigenvectors.

For the 2-by-2 matrix 2*I the characteristic polynomial is (x - 2)^2, while the minimal polynomial is just x - 2. This is the minimal polynomial of a matrix, not of an algebraic number: for a scalar a, it is x - a.

Input arguments

A: Square numeric or symbolic matrix.
x: Symbolic variable for the polynomial form.

Output arguments

p: Coefficients (row vector) or a polynomial in x.

Example

Compare with the characteristic polynomial

syms x
A = sym([2 0; 0 2]);
minpoly(A, x)
charpoly(A, x)
minpoly(sym([1 2; 3 4]), x)

Output

ans = x - 2
ans = x^2 - 4*x + 4
ans = x^2 - 5*x - 2

minpoly