Matrix Minimal Polynomial

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

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

then its minimal polynomial is given by

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

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

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References

Dummit, D. S. and Foote, R. M. Abstract Algebra, 3rd ed. Hoboken, NJ: Wiley, 2004.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 Resource. https://mathworld.wolfram.com/MatrixMinimalPolynomial.html

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