Page 252 - A Course in Linear Algebra with Applications

236 Chapter Seven: Orthogonality in Vector Spaces

Example 7.3.7
Find all real orthogonal 2 x 2 matrices.
Suppose that the real matrix

A=r \
c a

T
is orthogonal; thus A A = I 2. Equating the entries of the
T
matrix A A to those of I 2, we obtain the equations

2
a 2 + c 2 = 1 = b + d 2 and ab + cd = 0.

Now the first equation asserts that the point (a, c) lies on the
circle x 2 + y 2 = 1. Hence there is an angle 9 in the interval
[0, 2n] such that a = cos 9 and c = sin 9. Similarly there is
an angle 4> in this interval such that b = cos 0 and d = sin (p.
Now we still have to satisfy the third equation ab+cd = 0,
which requires that

cos 9 cos 4> + sin 9 sin 0 = 0

that is, cos(c/> - 9) = 0. Hence 4> - 9 = ±?r/2 or ±3TT/2. We
need to solve for b and d in each case. If <> = 0 + 7r/2 or
/
cp = 9 — 37r/2, we find that b — — sin 9 and d = cos 9. If, on
the other hand, (f) = 9 - n/2 or 0 = 9 + 37r/2, it follows that
b = sin 9 and d = — cos 0.
We conclude that >1 has of one of the forms

cos 9 sin 9
sin 9 — cos 0

with 9 in the interval [0, 2n\. Conversely, it is easy to verify
that such matrices are orthogonal. Thus the real orthogonal
2 x 2 matrices are exactly the matrices of the above types.