Fundamentals of Matrix Algebra, 2011a

Chapter 2
.
Matrix Arithmec
⎡ ⎤
1
2. ⃗v⃗y = [ 2 0 1 −1 ] ⎢ 2
⎥
⎣ 5 ⎦ = 2(1) + 0(2) + 1(5) − 1(0) = 7
0
3. ⃗u⃗y is not defined; Definion 13 specifies that in order to mulply a row vector
and column vector, they must have the same number of entries.
4. ⃗u⃗v is not defined; we only know how to mulpy row vectors by column vectors.
We haven’t defined how to mulply two row vectors (in general, it can’t
be done).
5. The product ⃗x⃗u is defined, but we don’t know how to do it yet. Right now, we
only know how to mulply a row vector mes a column vector; we don’t know
how to mulply a column vector mes a row vector. (That’s right: ⃗u⃗x ≠ ⃗x⃗u!)
Now that we understand how to mulply a row vector by a column vector, we are
ready to define matrix mulplicaon.
.
Ḋefinion 14
Matrix Mulplicaon
.
Let A be an m × r matrix, and let B be an r × n matrix. The
matrix product of A and B, denoted A·B, or simply AB, is the
m × n matrix M whose entry in the i th row and j th column is
the product of the i th row of A and the j th column of B.
It may help to illustrate it in this way. Let matrix A have rows ⃗a 1 , ⃗a 2 , · · · , ⃗a m and let
B have columns ⃗b 1 , ⃗b 2 , · · · , ⃗b n . Thus A looks like
⎡
⎤
− ⃗a 1 −
− ⃗a 2 −
⎢
⎣
⎥
. ⎦
− ⃗a m −
where the “−” symbols just serve as reminders that the ⃗a i represent rows, and B looks
like
⎡
| | |
⎤
⎣ ⃗b 1
⃗b 2 · · · ⃗b n
⎦
| | |
where again, the “|” symbols just remind us that the ⃗b i represent column vectors. Then
⎡
⎤
⃗a 1
⃗b 1 ⃗a 1
⃗b 2 · · · ⃗a 1
⃗b n
⃗a 2
⃗b 1 ⃗a 2
⃗b 2 · · · ⃗a 2
⃗b n
AB = ⎢
⎣
.
. . ..
. ⎥
⎦ .
⃗a m
⃗b 1 ⃗a m
⃗b 2 · · · ⃗a m
⃗b n
54