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moienMath 21b: Linear Algebra Spring 2016
Homework 14: Orthogonal transformations
This homework is due on Monday, March 7, respectively on Tuesday, March 8, 2016.
1 Determine from each of the following matrices whether they are
orthogonal:
1 1 1 1 1 1 1 1 1 0 0 0
1 1 1 1 1 −1 1 −1 0 −1 0 0
a) /2, b) /2, c)
,
1 1 1 1 1 1 −1 −1 0 0 −1 0
1 1 1 1 1 −1 −1 1 0 0 0 1
0 0 1 0 cos(1) sin(1) 0 0
0 −1 0 0 − sin(1) cos(1) 0 0
d) , e)
.
1 0 0 0 0 0 cos(2) sin(2)
0 0 0 1 0 0 sin(2) − cos(2)
2 If A, B are orthogonal, then
a) Is A + B is orthogonal? b) is 3A is orthogonal? c) Is AT is
orthogonal? d) B −1 is orthogonal? e) Is B −1AB orthogonal?
a b
3 a) Matrices of the form can be multiplied and the result
−b a
is of the same form. These rotation dilation matrices are also
called “complex numbers”! Which of these matrices plays the
√
role of i = −1, that is, which of them has the property that
A2 = −1 (where −1 means −I2)?
b) Figure out the formula for the multiplication
(a + ib)(c
+ id) of
a b c d
complex numbers by looking at the product .
−b a −d c
c) If you draw complex numbers a + ib, c + id as vectors, what is
the multiplication geometrically?
4 Mathematicians for a long time looked for higher dimensional ana-
logues of the complex numbers. Matrices of the form A(p, q, r, s) =
p −q −r −s
q p s −r
are called quaternions. They were invented
r −s p q
s r −q p
by Hamilton.
a) Find a basis for the set of all the matrices above.
b) Check that every matrix in the unit sphere p2 +q 2 +r2 +s2 = 1
in the four dimensional space of quaternions corresponds to an or-
thogonal matrix.
5 a) Explain why the identity matrix is the only n × n matrix that
is orthogonal, upper triangular and has positive entries on the
diagonal. b) Show that the QR factorization of an invertible
n × n matrix A is unique. That is, if A = Q1R1 and A = Q2R2
are two factorizations, argue why Q1 = Q2 and R1 = R2.
Orthogonal transformations
The transpose ATij = Aji satisfies (AB)T = B T AT and (AT )T =
A. The rank of the transpose is the same as the rank of A. An
n × n matrix A is orthogonal if AT A = 1 = 1n. The linear
transformation of an orthogonal matrix is called an orthog-
onal transformation. It preserves length and angle. The
column vectors of an orthogonal matrix forms an orthonormal
basis. The product of two orthogonal matrices is orthogonal.
The inverse A−1 is orthogonal and given by AT . Examples of
orthogonal transformations are rotations or reflections or the
identity matrix.