Page 58 - Matrices theory and applications

41
3. Matrices with Real or Complex Entries
∗
M M are Hermitian. We denote by H n the set of Hermitian matrices in
M n (CC). It is an IR-linear subspace of M n (CC), though it is not a CC-linear
subspace, since iM is skew-Hermitian when M is Hermitian.
∗
A square matrix M ∈ M n(CC)is saidtobe unitary if M M = I n .Since
this means that M is invertible, with inverse M , and since the left and
∗
= I n .The
∗
the right inverses are equal, an equivalent criterion is MM
set of unitary matrices in M n (CC) forms a multiplicative group, denoted
∗ 2
by U n . Unitary matrices satisfy | det M| =1, since det M M = | det M|
for every matrix M. The set of unitary matrices whose determinant equals
1, denoted by SU n is obviously a normal subgroup of U n . Finally, M is
∗
said to be normal if M and M commute: MM = M M. The Hermitian,
∗
∗
skew-Hermitian, and unitary matrices are normal.
Observe that the real orthogonal (respectively symmetric, skew-sym-
metric) matrices are unitary (respectively Hermitian, skew-Hermitian).
Conversely, if M is real and either unitary, symmetric, or skew-symmetric,
then M is either orthogonal, Hermitian, or skew-Hermitian.
A sesquilinear form on a complex vector space is a map
(x, y) → x, y
,
linear in x and satisfying
y, x
= x, y
.
It is thus antilinear in y:
¯
x, λy
= λ x, y
.
When y = x, x, y
= x, x
is a real number. The map x → x, x
is called
a Hermitian form. The correspondence between sesquilinear and Hermitian
formsisone-to-one.
Given a matrix M ∈ M n(CC), the form

(x, y) → m jk x j y k ,
j,k
n
n
defined on CC × CC , is sesquilinear if and only if M is Hermitian. It fol-
lows that there is an isomorphism between the sets of Hermitian matrices,
n
Hermitian, and sesquilinear forms on CC . As a matter of fact, a Hermitian
form can be written in the form

x → m jk x j ¯x k .
j,k
The kernel of a Hermitian or a sesquilinear form is the set of vectors
x ∈ E such that x, y
=0forevery y ∈ E. It equals the set of vectors
n
y ∈ E such that x, y
=0 for every x ∈ E.If E = CC ,itisalso the kernel
T
of M ,where M is the (Hermitian) matrix associated to the Hermitian
form. One says that the Hermitian form is degenerate if its kernel does not