Find the minimal polynomial of a matrix:

The corresponding matrix polynomial is zero:

The zero matrix has the simplest possible minimal polynomial:

The identity matrix has the next to simplest possible:

The factored form of the matrix minimal polynomial gives important information about the eigenspaces of the matrix. Namely, it indicates how much larger the generalized eigenspace for is than its ordinary eigenspace. Consider the follow matrix :

Compute its minimal polynomial :

And its characteristic polynomial :

Because has a linear term in , its generalized and ordinary eigenspaces are equal. The eigenspace is:

The geometric multiplicity of , , matches its algebraic multiplicity, the power of in :

Confirm the generalized eigenspace is the ordinary eigenspace by showing and have the same null spaces:

The presence of in means has a larger null space than :

The generalized eigenspace is now larger than the ordinary eigenspace:

Since the algebraic multiplicity of is , higher powers of have the same null space:

Finally, the presence of in means that has larger null spaces for and but not :

Use the minimal polynomial to compute the DrazinInverse:

The coefficients can be used to compute the Drazin inverse:

Verify some properties of the Drazin inverse:

Compare with the result of DrazinInverse:

The minimal polynomial of is the lowest-order polynomial for which yields a zero matrix:

Confirm that is an zero matrix:

Since is quintic, no quadric (or lower-order) polynomial can give zero when evaluated at :

The degree of a matrix minimal polynomial can be anywhere from 1 to the matrix dimension n:

This is different from the characteristic polynomial, which always has degree n:

The minimal polynomial divides the characteristic polynomial with a (possibly constant) polynomial quotient:

The minimal polynomial and the characteristic polynomial have the same distinct roots:

In particular, this means divides raised to the power:

A matrix minimal polynomial is often the same as the characteristic polynomial up to a constant factor:

Estimate the fraction of zero–one matrices for which this is true:

Compute a minimal polynomial:

The last element in the diagonal of the Smith normal form of a characteristic matrix gives the minimal polynomial:

Find the last block in the Frobenius normal form:

Extract the last column:

This gives the negatives of the minimal polynomial coefficients with lead coefficient implicitly set to 1:

The polynomial returned by MatrixMinimalPolynomial is generally monic for exact input:

However, it may not be monic for inexact input:

Make the polynomial monic: