Page 403 - Applied Numerical Methods Using MATLAB

392 MATRICES AND EIGENVALUES
which was solved by using Laplace transform in Problem P6.1. In this prob-
lem, we solve it again by the decoupling method through diagonalization of
the system matrix.
(a) Show that the eigenvalues and eigenvectors of the system matrix are as
follows.

1 1
λ 1 =−1, λ 2 =−2; v 1 = , v 2 = (P8.3.2)
−1 −2
(b) Show that the diagonalization of the above vector differential equation
v 2 ] yields the following equation:
using the modal matrix V = [ v 1

w 1 (t) −1 0 w 1 (t) 1
= + u s (t) (P8.3.3)

w 2 (t) 0 −2 w 2 (t) −1

−w 1 (t) + u s (t) w 1 (0) 2
= with =
−2w 2 (t) − u s (t) w 2 (0) −1
(c) Show that these equations can be solved individually by using Laplace
transform technique to yield the following solution, which is the same
as Eq. (P6.1.2) obtained in Problem P6.1(a).

1 1 −t
W 1 (s) = + , w 1 (t) = (1 + e )u s (t) (P8.3.4a)
s s + 1
−1/2 1/2 1 −2t
W 2 (s) = − , w 2 (t) =− (1 + e )u s (t) (P8.3.4b)
s s + 2 2
−t −2t
x 1 (t) 1/2 + e − (1/2)e
= u s (t) (P8.3.5)
x 2 (t) −e −t + e −2t
8.4 Householder Method and QR Factorization

This method can zero-out several elements in a column vector at each iter-
ation and make any N × N matrix a (lower) triangular matrix in (N − 1)
iterations.
(a) Householder Reflection (Fig. P8.4)
Show that the transformation matrix by which we can multiply a vector
x to generate another vector y having the same norm is

T
H = [I − 2ww ]
x − y 1
with w = = (x − y), c =||x − y|| 2 , ||x|| = ||y|| (P8.4.1)
||x − y|| 2 c

T
and that this is an orthonormal symmetric matrix such that H H =
HH = I; H −1 = H. Note the following facts.