Page 14 - Numerical Methods for Chemical Engineering

Review of scalar, vector, and matrix operations 3

differential equations, etc. Also, as we shall see, this solver fails for certain common
classes of very large systems of equations, and we need to know enough about linear
algebra to diagnose such situations and to propose other methods that do work in such
instances. This chapter therefore contains not only an explanation of how the MATLAB
solver is implemented, but also a detailed, fundamental discussion of the properties of linear
systems.
Our discussion is intended only to provide a foundation in linear algebra for the practice of
numerical computing, and is continued in Chapter 3 with a discussion of matrix eigenvalue
analysis. For a broader, more detailed, study of linear algebra, consult Strang (2003) or
Golub & van Loan (1996).

Review of scalar, vector, and matrix operations

As we use vector notation in our discussion of linear systems, a basic review of the concepts
of vectors and matrices is necessary.

Scalars, real and complex
Most often in basic mathematics, we work with scalars, i.e., single-valued numbers. These
may be real, such as 3, 1.4, 5/7, 3.14159 . . . . , or they may be complex, 1 + 2i, 1/2 i, where
√
i = −1. The set of all real scalars is denoted . The set of all complex scalars we call
C. For a complex number z ∈ C, we write z = a + ib, where a, b ∈ and

a = Re{z} = real part of z
b = Im{z} = imaginary part of z (1.8)

∗
The complex conjugate, ¯ z = z ,of z = a + ib is
∗
¯ z = z = a − ib (1.9)
Note that the product ¯ zz is always real and nonnegative,
2
2
2
2
2
¯ zz = (a − ib)(a + ib) = a − iab + iab − i b = a + b ≥ 0 (1.10)
so that we may define the real-valued, nonnegative modulus of z, |z|,as
√
2
2
|z|= ¯ zz = a + b ≥ 0 (1.11)
Often, we write complex numbers in polar notation,
−1
z = a + ib =|z|(cos θ + i sin θ) θ = tan (b/a) (1.12)
Using the important Euler formula, a proof of which is found in the supplemental material
found at the website that accompanies this book,
iθ
e = cos θ + i sin θ (1.13)