Page 19 - Numerical Methods for Chemical Engineering

8 1 Linear algebra

Matrix dimension
For a linear system Ax = b,
     
a 11 a 12 ... a 1N x 1 b 1
...
a 21 a 22 a 2N x 2 b 2
     
A =  . .      (1.42)

 . . . . .  x =  .  b =  . 
.
.
.
 . 
. 
 . 
a N1 a N2 ... a NN x N b N
to have a unique solution, there must be as many equations as unknowns, and so typically
A will have an equal number N of columns and rows and thus be a square matrix. A matrix
is said to be of dimension M × N if it has M rows and N columns. We now consider some
simple matrix operations.
Multiplication of an M × N matrix A by a scalar c
   
a 11 a 12 ... a 1N ca 11 ca 12 ... ca 1N
... ...
a 21 a 22 a 2N ca 21 ca 22 ca 2N
   

cA = c  . .   . . .  (1.43)
 . . . . .  =  . . . . . . 
.

. 

a M1 a M2 ... a MN ca M1 ca M2 ... ca MN
Addition of an M × N matrix A with an equal-sized M × N matrix B
   
a 11 ... a 1N b 11 ... b 1N
 a 21 ... a 2N   b 21 ... b 2N 


 . .  +  . 

 . . . .   . . . 
.
. 
a M1 ... a MN b M1 ... b MN
 
a 11 + b 11 ... a 1N + b 1N
a 21 + b 21 a 2N + b 2N
 ... 
 
. . (1.44)
. .
=  
 . . 
a M1 + b M1 ... a MN + b MN
Note that A + B = B + A and that two matrices can be added only if both the number of
rows and the number of columns are equal for each matrix. Also, c(A + B) = cA + cB.
Multiplication of a square N × N matrix A with an N-dimensional
vector v
This operation must be defined as follows if we are to have equivalence between the coef-
ficient and matrix/vector representations of a linear system:
     
a 11 a 12 ... a 1N v 1 a 11 v 1 + a 12 v 2 + ··· + a 1N v N
a 21 a 22 a 2N v 2 a 21 v 1 + a 22 v 2 + ··· + a 2N v N
 ...     

Av =  . .     . 
 . . . . .   .  =  . . 
.
.


.   . 
a N1 a N2 ... a NN v N a N1 v 1 + a N2 v 2 +· · · + a NN v N
(1.45)