LINEAR ALGEBRA 313
Definition B.0.5 (Orthogonality) An orthogonal real-valued matrix consists of columns
that are orthogonal to each other, i.e. the inner product between different columns equals
q
T
i
q
j
= 0. A matrix is termed orthonormal if its columns are orthogonal and additionally
have unit length
q
T
i
q
j
= δ(i, j) (B.9)
Orthonormal matrices have the properties
Q
T
Q = I
N
⇔ Q
T
= Q
−1
(B.10)
Definition B.0.6 (Unitary Matrix) An N × N matrix U with orthonormal columns and
complex elements is called unitary. The Hermitian of a unitary matrix is also its inverse
U
H
U = UU
H
= I
N
⇔ U
H
= U
−1
(B.11)
The columns of U span an N -dimensional orthonormal vector space.
Unitary matrices U have the following properties:
● The eigenvalues of U have unit magnitude (|λ
i
|=1).
● The eigenvectors belonging to different eigenvalues are orthogonal to each other.
● The inner product x
H
y between two vectors is invariant to multiplications with a
unitary matrix because (Ux)
H
(Uy) = x
H
U
H
Uy = x
H
y.
● The norm of a vector is invariant to the multiplication with a unitary matrix,
Ux=x.
● A random matrix B has the same statistical properties as the matrices BU and UB.
● The determinant of a unitary matrix equals det(U) = 1 (Blum, 2000).
Definition B.0.7 (Eigenvalue Problem) The calculation of the eigenvalues λ
i
and the asso-
ciated eigenvectors x
i
of a square N × N matrix A is called the eigenvalue problem. The
basic problem is to find a vector x proportional to the product Ax. The corresponding
equation
A · x = λ · x (B.12)
can be rewritten as
(
A − λ I
N
)
x = 0. For the non-trivial solution x = 0, the matrix
(
A − λ I
N
)
has to be singular, i.e. its columns are linear dependent. This results in
det
(
A − λ I
N
)
= 0 and the eigenvalues λ
i
represent the zeros of the characteristic poly-
nomial p
N
(λ) = det(A − λI
N
) of rank N . Each N × N matrix has exactly N eigenvalues
that need not be different.
In order to determine the associated eigenvectors x
i
, the equation
A − λ
i
I
N
x
i
= 0
has to be solved for each eigenvalue λ
i
. Since x
i
as well as c · x
i
fulfil the above equation,
a unique eigenvector is only obtained by normalising it to unit length. The eigenvectors
x
1
,...,x
k
belonging to different eigenvalues λ
1
,...,λ
k
are linear independent of each other
(Horn and Johnson, 1985; Strang, 1988).
The following relationships exist between the matrix A and its eigenvalues: