Page 61 - Matrices theory and applications

3. Matrices with Real or Complex Entries
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Theorem 3.1.1 Let n ∈ IN and let P ∈ CC[X] be a polynomial of degree
n,
n
P(X)= p 0 + p 1 X + ··· + p n X .
Let x be a root of P, with multiplicity µ,and let d be the distance from x to
the other roots of P.Let D be an open disk, D = D(x; ρ),with 0 <ρ <d.
Then there exists a number > 0 such that if Q ∈ CC[X] has degree n,
n
Q(X)= q 0 + q 1 X + ··· + q n X ,
and if
max |q j − p j | < ,
j
then D contains exactly µ roots of Q, counting multiplicities.
Let us apply this result to the characteristic polynomial of a given matrix.
Since the coefficients of the characteristic polynomial p M are polynomial
functions of the entries of M,the map M → p M is continuous from M n (CC)
to the set of polynomials of degree n.From Rouch´e’s theorem, we have the
following result.
Theorem 3.1.2 Let M ∈ M n (CC),and let λ be one of its eigenvalues, with
multiplicity µ,andlet d be the distance from λ to the other eigenvalues of
M.Let D be an open disk, D = D(λ; ρ),with 0 <ρ < d.Let us fix a norm
on M n(CC).
There exists an > 0 such that if A ∈ M n(CC) and A < ,the sum of
algebraic multiplicities of the eigenvalues of M + A in D equals µ.
Let us remark that this statement becomes false if one considers the
geometric multiplicities.
One often invokes this theorem by saying that the eigenvalues of a ma-
trix are continuous functions of its entries. Here is an interpretation. One
adapts the Hausdorff distance between compact sets so as to take into ac-
count the multiplicity of the eigenvalues. If M, N ∈ M n (CC), let us denote
by (λ 1 ,... ,λ n )and (θ 1 ,... ,θ n ) their eigenvalues, repeated according to
their multiplicities. One then defines
d(Sp M, Sp N):= inf max |λ j − θ σ(j) |,
j
σ∈S n
where S n is the group of permutations of the indices {1,... ,n}. This num-
ber is called the distance between the spectra of M and N.Withthis
notation, one may rewrite Theorem 3.1.2 in the following form.

Proposition 3.1.3 If M ∈ M n (CC) and α> 0,there exists > 0 such
that N − M < implies d(Sp M, Sp N) <α.

A useful consequence of Theorem 3.1.2 is the following.