Matrices
Definition of the sum A  B
2 matrices A and B can be added if they have the same shape.
Let A be an m x n matrix with elements aij
Let B be an m x n matrix with elements bij
Then we define A  B as an m x n matrix with element (aij + bij)
e.g.
3 0
1 2
A
B
1 7
3 4
Then A  B 
2 2
4 3
Definition of numerical multiple r A :
Let A be an m x n matrix with elements aij.
We defined r A as an m x n matrix with elements raij.
e.g.
3A 
9
0
3  21
By convention we write - A for (-1) A
And A - B for A +(-1) B
Properties of matrix addition and numerical multiplication
Let A , B and C be m x n matrices and r,s  
1. r A + s A = (r + s) A
2. r A + r B = r( A + B )
3. r(s A ) = (rs) A
4. A + B = B + A
5. ( A + B ) + C = A + ( B + C )
6. A + 0 = A
7. A + - A = 0
Matrices And Linear Form
An m x n matrix describes a function transformation from Rn to Rm
e.g. Transformation from R2 to R2
z1 = 3y1 – y2
R2  R2
z2 = 5y1 + 2y2
B(
3 1
)
5 2
A(
2 3 4
)
1 1 2
e.g. Transformation from R3 to R2
y1 = 2x1 + 3x2 + 4x3
y2 = x1 - x2 + 2x3
R3  R2
Transformation from R3 to R2 is described by
3.2  1.1 3.3  1.  1 3.4  1.2
C(
)  BA
5.2  2.1 5.3  2.  1 5.4  2.2
Some properties of matrix multiplication are
1. ( A  B)C  AC  BC
2. A( B  C )  AB  AC
3. (r A) B  r ( AB)  A(r B )
4. A( BC )  ( AB)C
5. 0 A  A0  0
6. IA  AI  A
Proof of A( BC )  ( AB)C
Let
A  p x q matrix
B  q x r matrix
AB is p x r matrix
C  r x s matrix
BC is q x s matrix
q
q
k 1
k 1
r
r
r
( A( BC )) ij   aik ( BC ) kj   aik  bki cij
i 1
q
(( AB)C ) ij   ( AB) ik C ij   aik bki cij
i 1
i 1 k 1
Q.E.D
Linear Functions
Let A denote a linear function from Rn to Rm that satisfies:
f ( x  y )  f ( x)  f ( y )
xy
f (x)  f ( x)
x
Theorem:
A function of form Rn to Rm is linear if and only there exists an m x n
matrix A such that f(x) = A x for all real values of x.
Proof:
Assume f is defined by f(x) = Ax where A is an m x n matrix.
f(x + y) = A (x + y) = A x + A y = f(x) + f(y)
 x  A(r x)   ( A x)  f ( x)
Assume f is a linear function from Rn to Rm. Let j denote the jth natural basis vector in
Rn. And let x be real.
n
x   x j e j  x1e1  x2 e2  ...  xn en
j 1
Then f ( x)  f ( x1e1  x2 e2  ...xn en )
If linear f ( x)  x1 f (e1 )  x2 f (e2 )  ...  xn f (en )
x1
f (e j )  R m  ( f (e1 ) f (e2 )... f (en )).
x2
...
xn
x1
f ( x)  ( f (e1 ). f (en )).
x2
...
xn
f ( x) 
a11
a12
... a1n x1
a 21
a 22
...
...
... a 2 n x 2
.  Ax
... ... ...
a m1
am2
a1 j
a2 j
Where f(j) =
...
a mj
... a mn x n
Hence we have shown that if Rn  Rm is a linear function then the associated matrix
is A , such that f(x)= A x with the jth column been equal to f(ej).
e.g. Consider the linear transformation R2  R2 that rotates anticlockwise by an angle
.
e2
sin  , cos 
e1
cos  , sin 
cos 
 sin 
Theorem
R
sin 
cos 
Let f be a linear function so Rn transforms to Rm
And let g be a linear function so Rm transforms to Rn
mxn
f Axy
nxm
Then g(f(x)) = BA x
g B yz
Proof g(f(x) = g( A x) = B ( A x)  ( BA ) x