Page 135 - Numerical methods for chemical engineering

Eigenvector matrix decomposition and basis sets 121

Special eigenvector properties of normal matrices
H
H
Normal matrices, AA = A A, have additional eigenvector properties.
Theorem EVN1 (spectral decomposition) If A is a normal matrix, it is possible to find a
complete orthonormal set of eigenvectors even if the matrix has eigenvalues of multiplicity
greater than 1; i.e. det(A − λI) = 0 has repeated roots. The matrix W whose columns are
these eigenvectors is unitary, and we can write A as
A = W W H = diag(λ 1 ,λ 2 ,...,λ N ) (3.108)

Proof We first write A as a Schur decomposition,

A = URU H (3.109)
Taking the Hermitian conjugate,
H H
H
H
A = (URU ) = UR U H (3.110)
we then form the two matrix products
H
H
H
H
H
AA = URU (UR U ) = URR U H
H
H
H
H
H
A A = UR U (URU ) = UR RU H (3.111)
H
H
H
H
For A to be normal, AA = A A, R must be normal as well, RR = R R.For
   ¯ 
R 11 R 12 R 13 ... R 1N R 11
¯
 R 22 R 23 ... R 2N   ¯ R 12 R 22 
   ¯ ¯ ¯ 

R 33 R 3N  R =  R 13 R 23 R 33 
 ...  H 
R = 
. .  . . . .
 . .   . . . . 
 . .  . . . . 
¯ ¯ ¯ ¯
R NN R 1N R 2N R 3N ... R NN
(3.112)
H
H
RR = R R only if R is diagonal. As R is similar to A,
 
λ 1
 λ 2 
R = =   (3.113)
 ... 
λ N
The Schur decomposition for a normal matrix is therefore
A = U U H (3.114)
Postmultiplication by U yields
AU = U (3.115)
The general form of the eigenvector decomposition (3.89) is AW = W , where W is a matrix
whose column vectors are eigenvectors of A. Therefore, for any normal matrix A,wecan
form a unitary matrix whose column vectors are eigenvectors to write A in Jordan normal
form,
A = W W H (3.116)