Page 124 - Numerical methods for chemical engineering

110 3 Matrix eigenvalue analysis

In general if P ≤ N is the number of distinct roots of (3.39), we write
p(λ) = det(A − λI) = (λ 1 − λ) (λ 2 − λ) m 2 ··· (λ P − λ) m P (3.41)
m 1
m k is the (algebraic) multiplicity of λ k , i.e., the number of times that it is repeated as a root
of (3.39). The multiplicities must sum to N,
m 1 + m 2 + ··· + m P = N (3.42)

From p(λ = 0), we find the determinant to be the product of the eigenvalues,
m 1 m 2
det(A) = λ λ ...λ m P (3.43)
1 2 P
It may also be shown that the trace equals the sum of the eigenvalues,
(3.44)
tr(A) = a 11 + a 22 +· · · + a NN = λ 1 + λ 2 +· · · + λ N
A proof of this result is found in the supplemental material in the accompanying website.

Eigenvalues and the existence/uniqueness properties
of linear systems

We now consider the existence and uniqueness properties of Ax = b from the viewpoint of
eigenvalue analysis. Let A be an N × N matrix, with the P distinct eigenvalues λ 1 , λ 2 ,...,
[2]
[1]
λ P for eigenvectors w , w ,..., w [P] ,
Aw [k] = λ k w [k] (3.45)

Let these P distinct eigenvalues be ordered by increasing modulus,
|λ 1 |≤|λ 2 |≤· · · ≤|λ P | (3.46)

We use the ≤ sign, even though the eigenvalues are distinct, because with complex eigen-
values we may have the distinct, but equal modulus, values

λ k = a + ib λ k+1 = a − ib a, b ∈ (3.47)
We now examine the effect of A on an eigenvector associated with λ 1 ,

Aw [1] = λ 1 w [1] (3.48)
If any eigenvalue is zero, it will be λ 1 , as we have ordered the eigenvalues by increasing
modulus. Also, if λ 1 = 0, A is singular, as
m 1 m 2
det(A) = λ λ ··· λ m P = 0 (3.49)
1 2 P
Thus, if λ 1 = 0, the null space of A is not empty and there exists some w ∈ K A , w = 0
such that Aw = 0. But, when λ 1 = 0, as
Aw [1] = λ 1 w [1] = 0 (3.50)

any eigenvector w [1] for λ 1 = 0 is in the null space of A and dim(K A ) = m 1 .