Page 123 - Numerical methods for chemical engineering

Multiplicity and formulas for the trace and determinant 109

unit length if desired. Consider the example
21 D = det(A) = (2)(3) − (1)(1) = 6 − 1 = 5

A = (3.34)
13 T = tr(A) = 2 + 3 = 5
with the eigenvalues
√
1 1 1 1 5 5
2
λ 1,2 = T ± T − 4D = (5) ± 25 − (4)(5) = ± (3.35)
2 2 2 2 2 2
An eigenvector w [1] for
√
5 5
λ 1 = +
2 2
must satisfy
) √ *
1 5
[1] [1] [1] [1]
(a 11 − λ 1 )w 1 + a 12 w 2 = 0 ⇒− + w 1 + w 2 = 0 (3.36)
2 2
[1] [1]
If we set w = 1, and compute w from (3.36), we obtain
1 2
√ *

) T
1 5
[1]
w = 1 + (3.37)
2 2
Above, we have selected a unique eigenvector by arbitrarily setting the value of one com-
ponent. More generally, we may select some vector v, say at random, and choose as a
normalization, v · w [ j] = 1. We replace row m of the matrix (A − λ j I) with the elements
of v to obtain a modified matrix (A − λ j I) ++ . We then obtain the eigenvector by solving
the linear system
(A − λ j I) ++ w [ j] = e [m] e [m] (3.38)
j = δ mj
Shortly, we consider numerical methods to compute λ k and w [k] without the need for
tedious analytical calculations.

Multiplicity and formulas for the trace and determinant

For an N × N matrix A, the characteristic polynomial
N N N

p(λ) = det(A − λI) = ... ε i 1 ,i 2 ,...,i N (a i 1 ,1 − λδ i 1 ,1 ) ... (a i N ,N − λδ i N ,N )
i 1 =1 i 2 =1 i N =1
(3.39)
is of degree N and thus has N roots, so an N × N matrix has N eigenvalues. However, these
may not all be distinct; i.e., we may have λ j = λ k for some j = k. As an example, consider
the matrix
 
2 3 4 2
p(λ) = (2 − λ) (1 − λ)
A =  0 2 5  (3.40)
λ 1 = 2 λ 2 = 2 λ 3 = 1
0 0 1