Page 20 - Numerical Methods for Chemical Engineering

Review of scalar, vector, and matrix operations 9

Av is also an N-dimensional vector, whose j th component is

N

(Av) j = a j1 v 1 + a j2 v 2 + ··· + a jN v N = a jk v k (1.46)
k=1

We compute (Av) j by summing a jk v k along rows of A and down the vector,

 

 
v k
⇒⇒ a jk ⇒⇒  
 

Multiplication of an M × N matrix A with an N-dimensional vector v
From the rule for forming Av, we see that the number of columns of A must equal the
dimension of v; however, we also can define Av when M = N,

     
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 M1 a M2 ... a MN v N a M1 v 1 + a M2 v 2 +· · · + a MN v N
(1.47)
M
N
If v ∈ , for an M × N matrix A, Av ∈ . Consider the following examples:
1 123 14
     
     
1 2 3 4 30 1
2 312 11
     
4 3 2 1    = 20  2 =   (1.48)
     
3 456 32
     
11 12 13 14 130 3
4 564 29
Note also that A(cv) = cAv and A(v + w) = Av + Aw.
Matrix transposition
T
We define for an M × N matrix A the transpose A to be the N × M matrix
T
  
a 11 a 12 ... a 1N a 11 a 21 ... a M1
a 21 a 22 a 2N a 12 a 22 a M2
 ...   ... 
T    
A =  . . .  =  . . (1.49)
 . . . . . .   . . . . . 
.
. 
a M1 a M2 ... a MN a 1N a 2N ... a NM