18.1. THE MATRIX OF A LINEAR TRANSFORMATION 379
Definition 18.1.17 Let A be a m× n matrix. Then AT is the n×m matrix definedas(AT)
i j ≡ A ji. In other words, the ith row becomes the ith column.
Example 18.1.18 Let A =
(1 4 −6−3 2 1
).Then AT =
1 −34 2−6 1
.
There is a fundamental theorem about how the transpose relates to multiplication.
Lemma 18.1.19 Let A be an m×n matrix and let B be a n× p matrix. Then
(AB)T = BT AT (18.13)
and if α and β are scalars,
(αA+βB)T = αAT +βBT (18.14)
Proof: From the definition,((AB)T
)i j= (AB) ji = ∑
kA jkBki = ∑
k
(BT )
ik
(AT )
k j =(BT AT )
i j
The proof of Formula 18.14 is left as an exercise and this proves the lemma.
Definition 18.1.20 An n×n matrix, A is said to be symmetric if A = AT . It is saidto be skew symmetric if A =−AT .
Example 18.1.21
2 1 31 5 −33 −3 7
is symmetric and
0 1 3−1 0 2−3 −2 0
is skew sym-
metric.
Example 18.1.22 Find AT B+CT where A =(
1 2),B =
(1 1
),C =
(1 21 1
)(
1 2)T ( 1 1
)+
(1 21 1
)T
=
(1 12 2
)+
(1 21 1
)T
=
(2 24 3
)Example 18.1.23 For F an m× n matrix, it is always possible to do the multiplicationFT F.
This is true because FT is n×m and F is m×n.Another important observation is the following which will be used frequently.
Proposition 18.1.24 Let A be an m×n matrix. Let B =(b1 · · · bp
)where each
bk is a column vector or n×1 matrix. Then AB is an m× p matrix and
AB =(
Ab1 · · · Abp)
so the kth column of AB is just Abk.