The minimal polynomial of a matrix is the monic polynomial in of smallest degree such that
The minimal polynomial divides any polynomial with and, in particular, it divides the characteristic
If the characteristic polynomial factors
then its minimal polynomial is given by
for some positive integers , where the satisfy .
For example, the characteristic polynomial of the zero matrix is , whiles its minimal polynomial is . However, the characteristic
polynomial and minimal polynomial of
are both .
The following Wolfram Language code will find the minimal polynomial for the square matrix
in the variable .
See alsoAlgebraic Number Minimal Polynomial
, Cayley-Hamilton Theorem
Field Minimal Polynomial
, Rational Canonical
Portions of this entry contributed by Todd
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ReferencesDummit, 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
Algebra I. New York: W. H. Freeman, 1985.
on Wolfram|AlphaMatrix 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