Page 59 - Matrices theory and applications

3. Matrices with Real or Complex Entries
42
n
reduce to {0}.When E = CC , this amounts to det M =0. One says that
the form is nondegenerate otherwise.
If both E and F are endowed with nondegenerate sequilinear forms ·, ·
E
∗
and ·, ·
F , respectively, and if u ∈L(E, F), one defines u by the formula
∗
∀x ∈ F, y ∈ E.
u (x),y
E = x, u(y)
F ,
The map u → u is an IR-isomorphism from L(E, F)onto L(F, E), and
∗
n
∗ ∗
¯ ∗
one has (λu) = λu ,(u ) = u.When E = CC and F = CC m are endowed
∗
with the canonical sesquilinear forms x 1 y 1 + ··· , the matrix associated
∗
to u is simply the Hermitian adjoint of the matrix associated to u.The
n
canonical Hermitian form over CC is positive definite: x, x
> 0if x =0. It
n
allows us to define a norm by x = x, x
. Identifying CC with column
√
vectors, one also defines X = X X if X ∈ M n×1 (CC). This norm will
∗
be denoted by · 2 in Chapter 4. A matrix is unitary if and only if it is
n
associated with an isometry of CC :
n
u(x) = x , ∀x ∈ CC .
More generally, let M be a Hermitian matrix and ·, ·
the form that it
n
defines on CC .One says that M is positive definite if x, x
> 0for ev-
n
ery x = 0. Again, x, x
is a norm on CC . We shall denote by HPD n
the set of the positive definite Hermitian matrices; it is an open cone in
H n . Its closure consists of the Hermitian matrices M that define a posi-
tive semidefinite Hermitian form over CC n ( x, x
≥ 0 for every x). They
are called positive semidefinite Hermitian matrices. One defines similarly,
among the real symmetric matrices, those that are positive definite, respec-
tively positive semidefinite. The positive definite real symmetric matrices
form an open cone in Sym (IR), denoted by SPD n .
n
The natural ordering on Hermitian forms induces an ordering on Hermi-
tian matrices. One writes H ≥ 0 n when the Hermitian form associated to H
takes nonnegative values. More generally, one writes H ≥ h if H − h ≥ 0 n .
We likewise define an ordering on real-valued symmetric matrices, referring
to the ordering on real-valued quadratic forms. 2
If U is unitary, the matrix U MU is similar to M.If M is Hermitian,
∗
∗
skew-Hermitian, normal, or unitary and if U is unitary, then U MU is still
Hermitian, skew-Hermitian, normal, or unitary.
2
We warn the reader that another, completely different, order still denoted by the
symbol ≥ will be defined in Chapter 5. This one will concern real-valued matrices that
are neither symmetric nor even square. One expects that the context is never ambiguous.