Minimal polynomial of a matrix

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.


linalg::minpoly(A, x)


linalg::minpoly(A, x) computes the minimal polynomial of the square matrix A in x, i.e., the monic polynomial of lowest degree annihilating the matrix A.

The minimal polynomial of A divides the characteristic polynomial of A, by Cayley-Hamilton theorem.

If the matrix is defined over Dom::Float, then due to numerical errors the computed polynomial can have a degree higher than the dimension of the matrix. In such cases, linalg::minpoly returns the value FAIL. See Example 3.

The component ring of A must be a field, i.e., a domain of category Cat::Field.


Example 1

We define the following matrix over the rational numbers:

A := Dom::Matrix(Dom::Rational)(
  [[0, 2, 0], [0, 0, 2], [2, 0, 0]]

The minimal polynomial of the matrix A in the variable x is then given by:

delete x:  linalg::minpoly(A, x)

In this case, the minimal polynomial is in fact equal to the characteristic polynomial of A:

linalg::charpoly(A, x)

Example 2

The minimal polynomial of the matrix:

B := matrix([[0, 1, 0], [0, 0, 0], [0, 0, 0]])

is a polynomial of degree 2:

m := linalg::minpoly(B, x)

The characteristic polynomial of B has degree 3 and is divided by the minimal polynomial of B:

p := linalg::charpoly(B, x)

p / m

Example 3

For the following example, MuPAD® is not able to compute the minimal polynomial, and thus FAIL is returned:

C := Dom::Matrix(Dom::Float)([
   [7, 0, 0, 0, 0], [1, 0, 0, 0, 0], [1, 2, 0, 0, 0], 
   [1, 2, 3, 0, 0], [1, 2, 3, 4, 7]

delete x:  linalg::minpoly(C, x)
Warning: Cannot compute the minimal polynomial. [linalg::minpoly]

In fact, for this example MuPAD is not able to check for zero equivalence during Gaussian elimination and therefore chose a wrong pivot element.

If you perform the computation over the coefficient domainDom::ExpressionField(normal) instead, then in most cases the minimal polynomial can be computed:

C := matrix([
   [7, 0, 0, 0, 0], [1, 0, 0, 0, 0], [1, 2, 0, 0, 0],
   [1, 2, 3, 0, 0], [1, 2, 3, 4, 7]

linalg::minpoly(C, x)

However, in general this problem regarding zero recognition cannot be avoided.



A square matrix of a domain of category Cat::Matrix


An indeterminate

Return Values

Polynomial of the domain Dom::DistributedPolynomial([x],R), where R is the component ring of A, or the value FAIL.

See Also

MuPAD Functions

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