,
Recall that an n x n matrix can be considered as defining a transformation of Rn (that is, a mapping from Rn to itself). Any such matrix transformation preserves the algebraic addition and scalar multiplication. Notice that we have been considering additional geometric notions of length and orthogonality. What can we say about transformations that preserve lengths?
Def. An n x n matrix Q is called orthogonal if the columns of Q form an orthonormal basis for Rn.
Note that the columns are required to be orthonormal, not just orthogonal. (Unfortunately the traditional term for such a matrix does not make that requirement clear!)
Example: Rotation matrices acting on R2 are orthogonal, since they are of the form
,
and for any choice of angle,
and
.
In fact, all 2x2 orthogonal matrices have either this form, or a similar one. Indeed, the requirement that the columns have length one forces the first column to have the form
for some angle, and then the second column must be a length one vector perpendicular to the first, and is therefore either
or
.
so all 2x2 orthogonal matrices are either rotations, or rotations combined with a reflection.
How do we recognize orthogonal matrices in general, and what other useful properties do they have?
Theorem
Let Q be an n x n matrix.
Then the following conditions are
equivalent.
(a) Q is orthogonal;
(b) Q is invertible
and Q-1
=
QT;
(c)
Q
u·
Q
v
= u·
v
for all u, v
in R
n;
(d) ||Q
u|| = ||u|| for all u
in
R
n.
To verify these statements, we will show that (a) implies (b), which implies (c), which implies (d), which implies (a). First, suppose that (a) is true. Then, since its columns are a basis for Rn , we know that Q is invertible. Moreover, we know that the (i,j) entry of Q T Q is the dot product of row i of Q T (which is the same as columni of Q) and columnj ofQ, so by the orthonormality conditions this entry is 1 if i = j and 0 otherwise. Thus Q T Q = I, and hence Q T =Q-1. Next, if we assume that (b) is true, then
Now, if we assume that (c) is true, then in particularQ u· Q u = u·u and (d) follows. Finally, if we assume that (d) is true, then for ej the j'th standard unit vector, we know that the j'th column of Q is given by Q e j. So
and all of the columns of Q have norm one. For i different from j, we have
and
Consequently Qei · Q e j = 0 for i different from j, and hence the columns of Q are orthogonal.
Notice that orthogonal matrices are exactly those which preserve lengths, when considered as transformations of Rn, and that they also preserve perpendicularity between pairs of vectors. In fact, if the general angle between two vectors is defined as the arccosine of the dot product of the unit vectors parallel to them, then orthogonal matrices preserve all angles between pairs of vectors. The most convenient fact, computationally, about orthogonal matrices is that their inverses are just their transposes.
Example
What else can we conclude about orthogonal matrices?
Theorem
Let Q and P be
n x n orthogonal matrices. Then
(a)
detQ = ±1;
(b) PQ
is an orthogonal matrix;
(c) Q-1
is an orthogonal matrix.
In order to see these results, note first that
so (detQ)2 = 1, and (a) follows. Next,
so PQ is orthogonal. Finally,
so Q-1 is also orthogonal.
Notice that the 2x2 orthogonal case treated above exhibits these properties quite explicitly. The (counterclockwise) rotation matrices are the orthogonal matrices of determinant 1. The matrices corresponding to reflection followed by rotation are the orthogonal matrices of determinant -1. The product and inverse of rotations, or combinations of reflection and rotation, are again matrices of the same type.
The orthogonal matrices of determinant 1 have a further subtle property: they preserve orientation: while reflections interchange right-handed and left-handed images, rotations preserve the "handedness" of an image (or a basis).
