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Lecture 5: Matrix Algebra
Matrix additions and Scalar Multiplications
DeÖnition 5.1. Let A = [~a 1 ;~a 2 ; :::;~an] mn
= [aij ] mn
and B =
h ~b 1 ;
~b 2 ; :::;
~b n
i
mn
[bij ] mn
be two matrices of the same dimension, where columns
~ai =
a 1 i
a 2 i
. . .
ami
; ~bi =
b 1 i
b 2 i
. . .
bmi
2 R
m
and be a constant (scalar). In the short notation for matrix, A = [aij ]mn ; aij stands for
the entry in the i th row and j th columns. Then
A + B =
h
~a 1 + ~b 1 ;~a 2 + ~b 2 ; :::;~an + ~bn
i
mn
= [aij + bij ] mn
A = [~a 1 ; ~a 2 ; :::; ~an] mn
= [aij ] mn
Or in the expanded form,
a 11 a 12 ::: a 1 n
a 21 a 22 ::: a 2 n
. . .
am 1 ::: ::: amn
b 11 b 12 ::: b 1 n
b 21 b 22 ::: a 2 n
. . .
bm 1 ::: ::: bmn
a 11 + b 11 a 12 + b 12 ::: a 1 n + b 1 n
a 21 + b 21 a 22 + b 22 :::
am 1 + bm 1 ::: ::: amn + bmn
a 11 ::: a 1 n
. .
. :::
am 1 ::: amn
a 11 ::: a 1 n
. .
. :::
am 1 ::: amn
Example 5.1 Let
A =
5 ; B =
5 ; C =
Then
A + B =
3 A =
A B =
How about
A + C =?
Properties:
A + B = B + A
(A + B) + C = A + (B + C)
(A + B) = A + B:
Matrix Multiplications
DeÖnition. Let A = [~a 1 ; ~a 2 ; :::;~ap] mp
= [aij ] mp
and B =
h ~b 1 ;
~b 2 ; :::;
~b n
i
pn
= [bij ] pn
be two matrices, where columns
~ai =
a 1 i
a 2 i
. . .
ami
2 R
m ; i = 1; 2 ; :::; p;
bj =
b 1 j
b 2 j
. . .
bpj
2 R
p ;
respectively. Note that
number of columns in A = p = number of rows in B:
Then the product of A and B is a m n matrix AB whose i th column vector is
A~bi:
In other words,
AB =
h
A~b 1 ; A~b 2 ; :::; A~bn
i
mn
= [cij ] mn
BA =
3 3
Note that AB and BA have di§erent dimensions. (d) Both ñA 2 2 ; B 2 2 ; and
AB =
2 2
BA =
2 2
Apparently, even if AB and BA have the same size 2 2 ,
AB 6 = BA:
Another way to look at this deÖnition of multiplication is to use the fact that
1 p matrix times p 1 matrix = 1 1 matrix
which is just a number, i.e.,
a 1 a 2 ::: ap
b 1
b 2
. . .
bp
p X
i=
aibi:
Rewrite
A =
a 11 a 12 ::: a 1 p
a 21 a 22 ::: a 2 p
. . .
am 1 ::: ::: amp
A 1
A 2
Am
where n 1 matrix
Ai =
ai 1 ai 2 ::: aip
is the i th row of A:
is called a row vector. Then, we have the following so-called "row-column rule"
AB =
A 1
A 2
Am
h ~ b 1 ;
b 2 ; :::;
bn
i
A 1
A 2
Am
~b 1 ;
A 1
A 2
Am
~b 2 ; :::;
A 1
A 2
Am
~b n
A 1 ~b 1 A 1 ~b 2 ::: A 1 ~bn
A 2 ~b 1 A 2 ~b 2 ::: A 2 ~bn
. ..
.. A
i
~b j in row^ i^ column^ j^
Am
b 1 Am
b 2 ::: Am
bn
h
Ai~bj
i
mn
One can easily verify that
Ai~bj =
ai 1 ai 2 ::: aip
b 1 j
b 2 j
. . .
bpj
p X
k=
aikbkj = cij :
Example 5.3. Use the row-column rule to calculate
2
DeÖnition 5.2. Let A = [aij ] mp
. The transpose of A; denoting by A
T ; is a matrix of
p m whose ith row is the ith column of the original matrix A. In other words, A
T is formed
then (Ai)
T is a column vectors in R
p
. The "row-column rule" becomes
AB =
A 1
A 2
Am
mp
h ~b 1 ;
~b 2 ; :::;
~b n
i
pn
(A 1 )
T ~b 1 (A 1 )
T ~b 2 ::: (A 1 )
T ~bn
(A 2 )
T ~b 1 (A 2 )
T ~b 2 ::: (A 2 )
T ~bn
(Am)
T ~b 1 (Am)
T ~b 2 ::: (Am)
T ~bn
mn
h
(Ai)
T ~bj
i
mn
DeÖnition 5.4. A square matrix A is called symmetric if A = A
T :
Following matrices are symmetric:
A =
; B =
5 ; C =
The matrix
D =
is not symmetric.
Properties of Matrix Multiplication:
Let A; B; C are matrices whose sizes may vary and will be indicated by subscripts,
be a constant. The m m matrix Im whose diagonal entries are 1 and all other entries are
zero, i.e.,
Im =
mm
is called m m identity matrix. We have the following properties:
- A (BC) = (AB) C () Amp (BpnCnr) pr
= (AmpBpn) mn
Cnr
- A (B + C) = AB + AC () Amp (Bpn + Cpn) = AmpBpn + AmpCpn
- (B + C) A = BA + CA () (Bpn + Cpn) Anm = BpnAnm + CpnAnm
- (AB) = (A) B = A (B) () (AmpBpn) = (Amp) Bpn = Amp (Bpn)
- IA = AI = A () ImAmp = AmpIp = Amp
6. (A)
T = A
T
A
T
T
= A
8. (B + C)
T = B
T
T () (Bpn + Cpn)
T = (Bpn)
T
T
9. (BC)
T = C
T B
T () (BpnCnr)
T = (Cnr)
T (Bpn)
T
- If A is symmetric, so is A
T :
Example 5.5 (a) Let
A =
; B =
Then
A
2
Note that A 6 = 0; but A
2 = 0: So in matrix multiplication, there is no "cancellation rule".
AB =
BA =
Note that AB 6 = BA: This shows that the commutative rule doesnít hold for matrix multi-
plication.
B
T A
T
= (AB)
T :
Homework 5
- Find A + 2B; 3 A 3 B if
A =
; B =
- Given
A =
; B =
; C =
