Page 63 - Matrices theory and applications

3. Matrices with Real or Complex Entries
46
Theorem 3.2.1 If K = CC, the normal matrices are diagonalizable, using
unitary matrices:
−1
∗
∗
(M M = MM )=⇒ (∃U ∈ U n ;
M = U
diag(d 1 ,... ,d n )U).
Again, one says that normal matrices are unitarily diagonalizable.This
theorem contains the following properties.
Corollary 3.2.1 Unitary, Hermitian, and skew-Hermitian matrices are
unitarily diagonalizable.
Observe that among normal matrices one distinguishes each of the above
families by the nature of their eigenvalues. Those of unitary matrices have
modulus one, while those of Hermitian matrices are real. Finally, those of
skew-Hermitian matrices are purely imaginary.
Proof
We proceed by induction on the size n of the matrix M.If n =0, there
is nothing to prove. Otherwise, if n ≥ 1, there exists an eigenpair (λ, x):
Mx = λx, x 2 =1.
∗ ¯
Since M is normal, M−λI n is,too.Fromabove, wesee that (M −λ)x 2 =
¯
(M − λ)x 2 = 0, and hence M x = λx.Let V be a unitary matrix such
∗
1
∗
that V e = x. Then the matrix M 1 := V MV is normal and satisfies
∗ 1
1
1
¯ 1
M 1 e = λe . Hence it satisfies M e = λe . This amounts to saying that
1
M 1 is block-diagonal, of the form M 1 = diag(λ, M ). Obviously, M inherits

the normality of M 1. From the induction hypothesis, M , and therefore M 1
and M, are unitarily diagonalizable.
One observes that the same matrix U diagonalizes M , because M =
∗
U −1 DU implies M = U D U −1∗ = U −1 D U,since U is unitary.
∗
∗
∗
∗
Let us consider the case of a positive semidefinite Hermitian matrix H.If
2
HX = λX,then 0 ≤ X HX = λ X . The eigenvalues are thus nonnega-
∗
tive. Let λ 1 ,... ,λ p be the nonzero eigenvalues of H.Then H is unitarily
similar to
D := diag(λ 1 ,... ,λ p , 0,... , 0).
From this, we conclude that rk H = p.Let U ∈ U n be such that H =
√
UDU . Defining the vectors X α = λ α U α ,where the U α are the columns
∗
of U, we obtain the following statement.
Proposition 3.2.1 Let H ∈ M n (CC) be a positive semidefinite Hermitian
matrix. Let p be its rank. Then H has p real, positive eigenvalues, while the
eigenvalue λ =0 has multiplicity n − p.There exist p column vectors X α ,
pairwise orthogonal, such that
∗
∗
H = X 1 X + ··· + X p X .
p
1
Finally, H is positive definite if and only if p = n (in which case, λ =0 is
not an eigenvalue).