Page 384 - Applied Numerical Methods Using MATLAB

SIMILARITY TRANSFORMATION AND DIAGONALIZATION 373

%nm811 to get the eigenvalues & eigenvectors of a matrix A.
clear
A = [0 1;0 -1];
[V,L] = eig(A) %V = modal matrix composed of eigenvectors
% L = diagonal matrix with eigenvalues on its diagonal
e = eig(A), roots(poly(A)) %just for eigenvalues
L = V^ - 1*A*V %diagonalize through similarity transformation
% into a diagonal matrix having the eigenvalues on diagonal.

8.2 SIMILARITY TRANSFORMATION AND DIAGONALIZATION

Premultiplying a matrix A by P −1 and post-multiplying it by P makes a similarity
transformation
A → P −1 AP (8.2.1)
Remark 8.1 tells us how a similarity transformation affects the eigenval-
ues/eigenvectors.
Remark 8.1. Effect of Similarity Transformation on Eigenvalues/Eigenvectors

1. The eigenvalues are not changed by a similarity transformation.
−1 −1 −1 −1
|P AP − λI|=|P AP − P λIP |=|P ||A − λI||P |= |A − λI|
(8.2.2)
2. Substituting v = P w into Eq. (8.1.1) yields

−1
Av = λv, AP w = λP w = Pλw, [P AP ]w = λw
This implies that the matrix P −1 AP obtained by a similarity transformation
has w = P −1 v as its eigenvector if v is an eigenvector of the matrix A.
In order to understand the diagonalization of a matrix into a diagonal matrix
(having its eigenvalues on the main diagonal) through a similarity transformation,
we have to know the following theorem:

Theorem 8.1. Distinct Eigenvalues and Independent Eigenvectors.
If the eigenvalues of a matrix A are all distinct—that is, different from each
other—then the corresponding eigenvectors are independent of each other and,
consequently, the modal matrix composed of the eigenvectors as columns is
nonsingular.

Now, for an N × N matrix A whose eigenvalues are all distinct, let us put all
of the equations (8.1.1) for each eigenvalue-eigenvector pair together to write
 
0 · 0
λ 1
0 · 0
 λ 2 
A[v 1 v 2 ·· · v N ] = [v 1 v 2 ·· · v N ]   , AV = V
 · · · · 
0 0 · λ N
(8.2.3)