Page 44 - Matrices theory and applications

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2.5. The Characteristic Polynomial
Let us multiply these rows by the powers of M, beginning with M
0
ending with M
= I n . Summing the obtained equalities, we obtain the
expected formula.
For example, every 2 × 2 matrix satisfies the identity
2
M − (Tr M)M +(det M)I 2 =0.
2.5.1 The Minimal Polynomial n and
For a square matrix M ∈ M n (K), let us denote by J M the set of polyno-
mials Q ∈ K[X]such that Q(M) = 0. It is clearly an ideal of K[X]. Since
K[X] is Euclidean, hence principal (see Sections 6.1.1 and 6.1.2), there ex-
ists a polynomial Q M such that J M = K[X]Q M .In other words, Q(M)=0
and Q ∈ K[X]imply Q M |Q. Theorem 2.5.1 shows that the ideal J M does
not reduce to {0}, because it contains the characteristic polynomial. Hence,
Q M = 0 and one may choose it monic. This choice determines Q M in a
unique way, and one calls it the minimal polynomial of M. It divides the
characteristic polynomial.
Contrary to the case of the characteristic polynomial, it is not immedi-
ate that the minimal polynomial is independent of the field in which one
considers J M (note that we consider only fields that contain the entries of
M). We shall see in Section 6.3.2 that if L is a field containing K, then the
minimal polynomials of M in K[X]and L[X] are the same. This explains
the terminology.
Two similar matrices obviously have the same minimal polynomial, since

Q(P −1 MP)= P −1 Q(M)P.

If λ is an eigenvalue of M, associated to an eigenvector X,and if q ∈
K[X], then q(λ)X = q(M)X. Applied to the minimal polynomial, this
equality shows that the minimal polynomial is divisible by X − λ. Hence,
¯
if P M splits over K in the form
r

n j
(X − λ j ) ,
j=1
the λ j all being distinct, then the minimal polynomial can be written as
r

(X − λ j ) m j ,
j=1
with 1 ≤ m j ≤ n j . In particular, if every eigenvalue of M is simple, the
minimal polynomial and the characteristic polynomial are equal.
An eigenvalue is called semi-simple if it is a simple root of the minimal
polynomial.