522 Chapter 18Matrices and linear transformations
where
(18.58)
The transpose of Ais
(18.59)
where, for example, the row vectora
T1 = 1 (a
1a
2a
3)is the transpose of the column
vector a. The productA
TAofAand its transpose can then by written
(18.60)
where, for example,
(18.61)
(18.62)
We recognizea
Tbas the scalar producta 1
·1 bof the vectorsa 1 = 1 (a
1, a
2, a
3)and
b 1 = 1 (b
1, b
2, b
3)(see Section 16.5), and these vectors are orthogonal ifa
Tb 1 = 1 a 1
·
1 b 1 = 10.
Also, the quantitya
Tais the square of the length of the vector a, and the vector has
unit length ifa
Ta 1 = 1 |a|
21 = 11. It follows that when the columns of Aform a system of
orthonormal vectors, the productA
TAis the unit matrix:
AA (18.63)
aa ab ac
ba bb bc
ca cb cc
TTTTTTTTTT=
==
=
100
010
001
I
ab
T==+
()aaa
b
b
b
ab a
12312311 222 33bab+
aa
T==+
()aaa
a
a
a
aa
1231231222232+a
AA
a
b
c
abc
aa ab ac
TTTTTTT==
()bba bb bc
ca cb cc
TTTTTT
A
a
b
TT==
aaa
bbb
ccc
1
23123123TTTc
ab=, =
a
a
a
b
b
b
123123,=c
c
c
c
123