18.6 Orthogonal matrices and orthogonal transformations 521
A nonsingular transformation Afollowed by its inverse transformationA
− 1(orA
− 1followed by A) is therefore equivalent to the identity transformation I; that is, the
‘transformation’ that leaves every vector unchanged. Thus
if x′ 1 = 1 Ax and x′′ 1 = 1 A
− 1x′ then x′′ 1 = 1 A
− 1Ax 1 = 1 x (18.54)
EXAMPLE 18.21The matrix
is nonsingular, with determinant det 1 A 1 = 1 cos
21 θ 1 + 1 sin
2θ 1 = 11 , and represents rotation
through angleθ. The corresponding inverse transformation is the rotation through
angle−θ, with matrix
andB 1 = 1 A
− 1is the inverse matrix of A. Thus
In this particular case, the inverse is also equal to the transpose,A
− 11 = 1 A
T18.6 Orthogonal matrices and orthogonal transformations
A nonsingular square matrix is called orthogonalwhen its inverse is equal to its
transpose,
A
− 11 = 1 A
T(orthogonal matrix) (18.55)
For example, the matrix discussed in Example 18.21 is orthogonal:
(18.56)
The characteristic property of an orthogonal matrix is that its columns (and its rows)
form a system of orthogonal unit vectors (orthonormal vectors). For order 3, let
Aa==bc(18.57)
abc
abc
abc
1
11222333())
AA=
−
,=
−
cos sin
sin cos
cos sin
sin
θθ
θθ
θθ
Tθθθcos
=
−A
1BA=
−
cos sin −
sin cos
cos sin
sin co
θθ
θθ
θθ
θ ssθ
=
=
10
01
I
B=
−−−
−−
=
cos( ) sin( )
sin( ) cos( )
θθcos
θθ
θθθ
θθ
sin
−sin cos
A=
−
cos sin
sin cos
θθ
θθ