Page 18 - Numerical Methods for Chemical Engineering

Review of scalar, vector, and matrix operations 7

Table 1.1 p-norm values for the 3-D
vector (1, −2, 3)

p v p
1 6
√
2 14 = 3.742
10 3.005
50 3.00000000009

The length of a vector is thus also the 2-norm.For v = [1 −2 3], the values of the p-norm,
computed from (1.35), are presented in Table 1.1.

p
p 1/p
p
p
p 1/p
p
v p = [|1| +|−2| +|3| ] = [(1) + (2) + (3) ] (1.35)
We define the infinity norm as the limit of v p as p →∞, which merely extracts from v
the largest magnitude of any component,
v ∞ ≡ lim v p = max j∈[1,N] {|v j |} (1.36)
p→∞
For v = [1 −2 3], v ∞ = 3.
Like scalars, vectors can be complex. We define the set of complex N-dimensional vectors
N
N
as C , and write each component of v ∈ C as
√
v j = a j + ib j a j , b j ∈ i = −1 (1.37)
N
*
The complex conjugate of v ∈ C , written as ¯v or v ,is
∗
  
a 1 + ib 1 a 1 − ib 1
a 2 + ib 2 a 2 − ib 2
   
∗     (1.38)
.  =  . 
. .
v = 
 .   . 
a N + ib N a N − ib N
N
For complex vectors v, w ∈ C , to form the dot product v · w, we take the complex
conjugates of the first vector’s components,
N

∗
v · w = v w k (1.39)
k
k=1
This ensures that the length of any v ∈ C is always real and nonnegative,
N N N
2 ∗ 2 2
v = v v k = (a k − ib k )(a k + ib k ) = a + b ≥ 0 (1.40)
2 k k k
k=1 k=1 k=1
N
For v, w ∈ C , the order of the arguments is significant,
∗
v · w = (w · v) = w · v (1.41)