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Matrix Minimal Polynomial


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

 p(A)=sum_(i=0)^nc_iA^i=0.
(1)

The minimal polynomial divides any polynomial q with q(A)=0 and, in particular, it divides the characteristic polynomial.

If the characteristic polynomial factors as

 char(A)(x)=(x-lambda_1)^(n_1)...(x-lambda_k)^(n_k),
(2)

then its minimal polynomial is given by

 p(x)=(x-lambda_1)^(m_1)...(x-lambda_k)^(m_k)
(3)

for some positive integers m_i, where the m_i satisfy 1<=m_i<=n_i.

For example, the characteristic polynomial of the n×n zero matrix is (-1)^nx^n, whiles its minimal polynomial is x. However, the characteristic polynomial and minimal polynomial of

 [0 1; 0 0]
(4)

are both x^2.

The following Wolfram Language code will find the minimal polynomial for the square matrix a in the variable x.

  MatrixMinimalPolynomial[a_List?MatrixQ,x_]:=Module[
    {
      i,
      n=1,
      qu={},
      mnm={Flatten[IdentityMatrix[Length[a]]]}
    },
    While[Length[qu]==0,
      AppendTo[mnm,Flatten[MatrixPower[a,n]]];
      qu=NullSpace[Transpose[mnm]];
      n++
    ];
    First[qu].Table[x^i,{i,0,n-1}]
  ]

See also

Algebraic Number Minimal Polynomial, Cayley-Hamilton Theorem, Characteristic Polynomial, Extension Field Minimal Polynomial, Rational Canonical Form

Portions of this entry contributed by Todd Rowland

Explore with Wolfram|Alpha

References

Dummit, D. and Foote, R. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1991.Herstein, I. §6.7 in Topics in Algebra, 2nd ed. New York: Wiley, 1975.Jacobson, N. §3.10 in Basic Algebra I. New York: W. H. Freeman, 1985.

Referenced on Wolfram|Alpha

Matrix Minimal Polynomial

Cite this as:

Rowland, Todd and Weisstein, Eric W. "Matrix Minimal Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MatrixMinimalPolynomial.html

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