i
i
i
i
i
i
i
i
904 A. Some Linear Algebra
Some theoretical results of great use are: i)tr(A)=
n−1
i=0
a
ii
=
n−1
i=0
λ
i
, ii)det(A)=
.
n−1
i=0
λ
i
, iii)ifA is real (consists of only
real values) and is symmetric, i.e., A = A
T
, then its eigenvalues are real
and the different eigenvectors are orthogonal.
Orthogonal Matrices
Here, we will shed some light on the concept of an orthogonal matrix, its
properties, and its characteristics. A square matrix, M,withonlyreal
elements is orthogonal if and only if MM
T
= M
T
M = I.Thatis,when
multiplied by its transpose, it yields the identity matrix.
The orthogonality of a matrix, M, has some significant implications
such as: i) |M| = ±1, ii) M
−1
= M
T
, iii) M
T
is also orthogonal,
iv) ||Mu|| = ||u||, v) Mu ⊥ Mv ⇔ u ⊥ v, vi)ifM and N are
orthogonal, so is MN.
The standard basis is orthonormal because the basis vectors are mutu-
ally orthogonal and of length one. Using the standard basis as a matrix,
we can show that the matrix is orthogonal:
8
E =(e
x
e
y
e
z
)=I,and
I
T
I = I.
To clear up some possible confusion, an orthogonal matrix is not the
same as an orthogonal vector set (basis). Informally, the difference is that
an orthogonal matrix must consist of normalized vectors. A set of vectors
may be mutually perpendicular, and so be called an orthogonal vector set.
However, if these are inserted into a matrix either as rows or columns,
this does not automatically make the matrix orthogonal, if the vectors
themselves are not normalized. An orthonormal vector set (basis), on the
other hand, always forms an orthogonal matrix if the vectors are inserted
into the matrix as a set of rows or columns. A better term for an orthogonal
matrix would be an orthonormal matrix, since it is always composed of an
orthonormal basis, but even mathematics is not always logical.
A.3.2 Change of Base
Assume we have a vector, v, in the standard basis (see Section 1.2.1),
described by the coordinate axes e
x
, e
y
,ande
z
.Furthermore,wehave
another coordinate system described by the arbitrary basis vectors f
x
, f
y
,
and f
z
(which must be noncoplanar, i.e., |f
x
f
y
f
z
| =0). Howcanv be
expressed uniquely in the basis described by f
x
, f
y
,andf
z
? The solution is
given below, where w is v expressed in the new basis, described by F [741]:
Fw =
f
x
f
y
f
z
w = v
⇐⇒
w = F
−1
v.
(A.42)
8
Note that the basis is orthonormal, but the matrix is orthogonal, though they mean
thesamething.