Linear Algebra, 2020a

Section IV. Matrix Operations 237
Distributing and regrouping on the v’s gives
⎛
⎜
(1 · 4 + 1 · 5)v 1 +(1 · 6 + 1 · 7)v 2 +(1 · 8 + 1 · 9)v 3 +(1 · 2 + 1 · 3)v ⎞
4
⎟
= ⎝(0 · 4 + 1 · 5)v 1 +(0 · 6 + 1 · 7)v 2 +(0 · 8 + 1 · 9)v 3 +(0 · 2 + 1 · 3)v 4 ⎠
(1 · 4 + 0 · 5)v 1 +(1 · 6 + 0 · 7)v 2 +(1 · 8 + 0 · 9)v 3 +(1 · 2 + 0 · 3)v 4
which is this matrix-vector product.
⎛
⎜
1 · 4 + 1 · 5 1· 6 + 1 · 7 1· 8 + 1 · 9 1· 2 + 1 · 3
⎞
⎟
= ⎝0 · 4 + 1 · 5 0· 6 + 1 · 7 0· 8 + 1 · 9 0· 2 + 1 · 3⎠
1 · 4 + 0 · 5 1· 6 + 0 · 7 1· 8 + 0 · 9 1· 2 + 0 · 3
B,D
⎛ ⎞
v 1
v 2
⎜ ⎟
⎝v 3 ⎠
v 4
The matrix representing g ◦ h has the rows of G combined with the columns of
H.
B
D
2.3 Definition The matrix-multiplicative product of the m×r matrix G and the
r×n matrix H is the m×n matrix P, where
p i,j = g i,1 h 1,j + g i,2 h 2,j + ···+ g i,r h r,j
so that the i, j-th entry of the product is the dot product of the i-th row of the
first matrix with the j-th column of the second.
⎛
⎞
⎛
⎞ h 1,j
⎛
⎞
.
··· h 2,j ···
GH = ⎜
⎝g i,1 g i,2 ··· g i,r
⎟
⎠ ⎜
⎝
⎟
. ⎠ = .
⎜
⎝··· p i,j ··· ⎟
⎠
.
.
h r,j
2.4 Example
⎛
⎜
2 0
⎞
⎛
( )
⎟ 1 3 ⎜
2 · 1 + 0 · 5 2· 3 + 0 · 7
⎞ ⎛ ⎞
2 6
⎟ ⎜ ⎟
⎝4 6⎠
= ⎝4 · 1 + 6 · 5 4· 3 + 6 · 7⎠ = ⎝34 54⎠
5 7
8 2
8 · 1 + 2 · 5 8· 3 + 2 · 7 18 38
2.5 Example Some products are not defined, such as the product of a 2×3 matrix
with a 2×2, because the number of columns in the first matrix must equal the
number of rows in the second. But the product of two n×n matrices is always
defined. Here are two 2×2’s.
( )( )
1 2 −1 0
3 4 2 −2
(
=
1 · (−1)+2 · 2 1· 0 + 2 · (−2)
3 · (−1)+4 · 2 3· 0 + 4 · (−2)
) (
=
3 −4
5 −8
)