Page 57 - Matrices theory and applications

3
Matrices with Real or Complex Entries

Definitions

A square matrix M ∈ M n (IR)is said to be normal if M and M T commute:
T
T
M M = MM . The real symmetric, skew-symmetric, and orthogonal
matrices are normal.
In considering matrices with complex entries, a useful operation is com-
¯
z
plex conjugation z → ¯. One denotes by M the matrix obtained from M
by conjugating the entries. We then define the Hermitian adjoint matrix 1
∗
M by
¯ T
∗
M := (M) = M .
T
∗
One therefore has m ∗ = m ji and det M = det M.The map M → M ∗
ij
is an anti-isomorphism, which means that it is antilinear (meaning that
¯
∗
(λM) = λM ) and satisfies, moreover, the product formula
∗
∗
(MN) = N M .
∗
∗
∗ −1
When a square matrix M ∈ M n (CC) is invertible, then (M ) =(M −1 ∗
) .
This matrix is sometimes denoted by M −∗ .
One says that a square matrix M ∈ M n (CC)is Hermitian if M = M and
∗
skew-Hermitian if M = −M.If M ∈ M n×m(CC), the matrices MM and
∗
∗
1
We warn the reader about the possible confusion between the adjoint and the Her-
mitian adjoint of a matrix. One may remark that the Hermitian adjoint is defined for
every rectangular matrix with complex entries, while the adjoint is defined for every
square matrix with entries in a commutative ring.