Last Updated：2025/12/31
Sentence
Each
root
of
the
minimal
polynomial
of
a
matrix
M
is
an
eigenvalue
of
M
and
a
root
of
its
characteristic
polynomial.
(A
root
of
the
minimal
polynomial
has
a
multiplicity
that
is
less
than
or
equal
to
the
multiplicity
of
the
same
root
in
the
characteristic
polynomial.
Thus
the
minimal
polynomial
divides
the
characteristic
polynomial.
Also,
any
root
of
the
characteristic
polynomial
is
also
a
root
of
the
minimal
polynomial,
so
the
two
kinds
of
polynomial
have
the
same
roots,
only
(possibly)
differing
in
their
multiplicities.)
Quizzes for review
Each root of the minimal polynomial of a matrix M is an eigenvalue of M and a root of its characteristic polynomial. (A root of the minimal polynomial has a multiplicity that is less than or equal to the multiplicity of the same root in the characteristic polynomial. Thus the minimal polynomial divides the characteristic polynomial. Also, any root of the characteristic polynomial is also a root of the minimal polynomial, so the two kinds of polynomial have the same roots, only (possibly) differing in their multiplicities.)
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