Page 42 - Matrices theory and applications

2.5. The Characteristic Polynomial
is called the trace of M and is denoted by Tr M. One has the trivial formula
that if N ∈ M n×m (K)and P ∈ M m×n (K), then
Tr(NP)= Tr(PN).
For square matrices, this identity also becomes
Tr[N, P]= 0. 25
Since P M possesses n roots in K, counting multiplicities, one sees that
a square matrix has always at least one eigenvalue, which, however, does
not necessarily belong to K. The multiplicity of λ as a root of P M is called
algebraic multiplicity of the eigenvalue λ.The geometric multiplicity of λ is
n
the dimension of ker(λI n −M)in K . The sum of the algebraic multiplicities
of the eigenvalues of M (considered in K)is n, the size of the matrix. An
eigenvalue of algebraic multiplicity one (that is, a simple root of P M )is
called simple.It is geometrically simple if its geometric multiplicity equals
one.
The characteristic polynomial is a similarity invariant, in the following
sense:
Proposition 2.5.1 If M and M are similar, then P M = P M .In
particular, det M =det M and Tr M =Tr M .

The proof is immediate. One deduces that the eigenvalues and their
algebraic multiplicities are similarity invariants. This is also true for the
geometric multiplicities, by a direct comparison of the kernel of λI n −
M and of λI n − M . Furthermore, the expression obtained above for the

characteristic polynomial provides the following result.
Proposition 2.5.2 The product of the eigenvalues of M (considered in
K), counted with their algebraic multiplicities, is det M.Their sum is Tr M.

Let µ be the geometric multiplicity of an eigenvalue λ of M.Let us choose
n
abasis γ of ker(λI n − M), andthena basis of β of K that completes γ.
Using the change-of-basis matrix from the canonical basis to β, one sees
that M is similar to a matrix M = P −1 MP,whose µ first columns have

the form

λI µ
.
0 n−µ,µ
µ
A direct calculation shows then that (X−λ) divides P M ,that is, P M .The
geometric multiplicity is thus less than or equal to the algebraic multiplicity.
The characteristic polynomials of M and M T are equal. Thus, M and
M T have the same eigenvalues. We shall show in Chapter 6 a much deeper
result, namely M and M T are similar.
The main result concerning the characteristic polynomial is the Cayley–
Hamilton theorem: