Page 223 - MATLAB an introduction with applications

208 ——— MATLAB: An Introduction with Applications

Here we have the first element of the vector [P ][a ] to be non zero since the sub-diagonal elements of the
1
1

column 1 of matrix []A is required to be zero.
Now choose the vector {w } such that it must fulfill the condition
1
(I – θ{w }{w } )[a ] = ± ||a || e 1 ...(4.15)
T
1 2
1
1
1
where e is a non-dimensional unit vector from Eq. (4.15).
1
[a ] – θ{w } = ± ||a || e 1
1
1
1 2
T
where θ is a constant and equal to e{w } {w }.
1
1
Let us set θ = 1.0, then we obtain
{w } = [a ] + sign (a ) ||a || e 1
1
11
1 2
1
From equation (4.14) we can get
{v } = {w } [A]
T
T
1
1

T
hence, []A = [A] – θ({w }{v } )
1
1
The factorization is explained in the example.
4.10 SYMMETRIC MATRIX EIGENVALUE PROBLEMS
The standard matrix eigenvalue problem is
Ax = λx ...(4.16)
where A is a given n × n matrix. The objective here is to find the scalar λ and the vector x.
Equation (4.16) can be rewritten as
( A −λ ) I x = 0
A non-trivial solution exists only if
A −λ = 0 ...(4.17)
I
Expansion of Eq.(4.17) gives the polynomial equation called the characteristic equation.
a λ+ a λ n− 1 + + n a n+ 1 = 0
n
... a λ +
2
1
which has the roots λ , i = 1, 2, …, n, called the eigenvalues of the matrix A. The solutions of x of (A – λ I)
i
i
i
x = 0 are known as eigenvectors.
4.11 JACOBI METHOD
Applying the transformation x = Px* in Eq.(4.16) where P is a non-singular matrix, we can write
−
−
1
1
P APx * =λ P Px *
or Ax =λ x * ...(4.18)
**
–1
where A* = P AP.
Matrices that have the same eigenvalues are deemed to be similar and the transformation between them is
called a similarity transformation. Diagonalizing A*, Eq.(4.18) can be written as