Page 60 - Matrices theory and applications

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3.1. Eigenvalues of Real- and Complex-Valued Matrices
3.1 Eigenvalues of Real- and Complex-Valued
Matrices
Since CC is algebraically closed, every complex-valued square matrix, and
every endomorphism of a CC-vector space of dimension n ≥ 1, possesses
eigenvalues. As a matter of fact, the characteristic polynomial has roots.
A real-valued square matrix may not have eigenvalues in IR, but it has at
least one in CC.If n is odd, M ∈ M n(IR) has at least a real eigenvalue,
because P M is real of odd degree.
Proposition 3.1.1 The eigenvalues of Hermitian matrices, as well as
those of real symmetric matrices, are real.
Proof
Let M ∈ M n(CC) be a Hermitian matrix and let λ be one of its eigen-
values. Let us choose an eigenvector X: MX = λX. Taking the Hermitian
¯
adjoint, we obtain X M = λX. Hence,
∗
¯
∗
∗
∗
∗
λX X = X (MX)= (X M)X = λX X,
or
¯
∗
(λ − λ)X X =0.
2 ¯
However X X = |x j | > 0. Therefore, we are left with λ−λ = 0. Hence
∗
j
λ is real.
We leave it to the reader to show, as an exercise, that the eigenvalues of
skew-Hermitian matrices are purely imaginary.
Proposition 3.1.2 The eigenvalues of the unitary matrices, as well as
those of real orthogonal matrices, are complex numbers of modulus one.
Proof
As before, if X is an eigenvector associated to λ,one has
2
2
2
∗
∗
∗
|λ| X =(λX) (λX)= (MX) MX = X M MX = X X = X ,
∗
∗
2
and therefore |λ| =1.
3.1.1 Continuity of Eigenvalues
One of the more delicate statements in the elementary theory of matrices
concerns the continuity of the eigenvalues. Though a proof might be pro-
vided througth explicit bounds, it is easier to use Rouch´e’s theorem about
holomorphic functions. We begin with a statement concerning polynomials,
that is a bit less precise than Rouch´e’s theorem.