Page 122 - Numerical Methods for Chemical Engineering

Estimating eigenvalues; Gershgorin’s theorem 111

Estimating eigenvalues; Gershgorin’s theorem

With the significant effort required to find the roots of

det(A − λI) = (λ 1 − λ) (λ 2 − λ) m 2 ··· (λ P − λ) m P (3.51)
m 1
it would be convenient if we could just look at a matrix and be able to “tell” what are
its eigenvalues. Unfortunately, we cannot do this in general, although for a few cases it is
possible. For triangular and diagonal matrices
   
U 11 U 12 U 1N L 11
U 22 U 2N L 21 L 22
 ...   
   
. L =  .
U =  .   . . . 
 · .  . . 
U NN L N1 L N2 ... L NN
(3.52)
 
D 11
D 22
 
 
.
D =  . 
 . 
D NN
the determinant equals the product of the elements along the diagonal,
(3.53)
det(U) = U 11 U 22 U 33 ... U NN det(L) = L 11 L 22 L 33 ··· L NN
Thus, the characteristic equation is already factored,
det(U − λI) = (U 11 − λ)(U 22 − λ)(U 33 − λ) ··· (U NN − λ) (3.54)
The eigenvalues of a triangular (diagonal) matrix lie along the diagonal
... (3.55)
λ 1 = U 11 λ 2 = U 22 λ N = U NN
For matrices that are not triangular, we cannot determine the eigenvalues by inspection,
but we can obtain upper and lower bounds using Gershgorin’s theorem. Let A be an N × N
matrix,
 
a 11 a 12 a 13 ... a 1N
 a 21 a 22 a 23 ... a 2N 
 
a 

A =  31 a 32 a 33 ... a 3N  (3.56)
 . . . . 
 . . . . . . . . 
a N1 a N2 a N3 ... a NN
In row k, the diagonal element is a kk , and the sum of the magnitudes of the off-diagonal
elements is

k =|a k1 |+|a k2 |+· · ·+|a k,k−1 |+|a k,k+1 |+· · ·+|a kN | (3.57)
As the eigenvalues of A are, in general, complex, we make a graph of the complex plane and
place the eigenvalue λ = a + ib at (a, b) (Figure 3.2). On this graph, we add a circle for each
row k of the matrix. The center of the circle is placed at the location of the diagonal element
a kk in the complex plane, and the radius of the circle is the sum of the magnitudes of the