HP 15c - Manual (Page 130)

130 Section 4: Using Matrix Operations
130
Keystrokes Display
⋮
3
OA 3.0000 Enters a
32
.
4
OA 4.0000 Enters a
33
.
A 0.8660 Calculates ratio－too large.
¦ 0.2304 Again, too large.
¦ 0.1039 Again, too large.
¦ 0.0060 Again, too large.
¦ 3.0463 -05 Again, too large.
¦ 5.8257 -10 Negligible ratio.
lA -0.8730
Recalls a
11
=λ
1
.
lA -9.0006 -10 Recalls a
12
.
lA -2.0637 -09 Recalls a
13
.
lA -9.0006 -10 Recalls a
21
.
lA 9.3429 -11 Recalls a
22
=λ
2
.
lA 1.0725 -09 Recalls a
23
.
lA -2.0637 -09 Recalls a
31
.
lA 1.0725 -09 Recalls a
32
.
lA 6.8730 Recalls a
33
=λ
3
.
´U 6.8730 Deactivates User mode.
In the new coordinate system the equation of the quadratic surface is approximately
−0.8730 2
1
z + 0 2
2
z + 6.8730 2
3
z = 7
This is the equation of a hyperbolic cylinder.
Eigenvectors of a Symmetric Real Matrix
As discussed in the previous application, a real symmetric matrix A has real eigenvalues λ
1
,
λ
2
... and corresponding orthogonal eigenvectors q
l
, q
2
, ... .
This program uses inverse iteration to calculate an eigenvector q
k
that corresponds to the
eigenvalue λ
k
such that ||q
k
||
R
= 1. The technique uses an initial vector z
(0)
to calculate
subsequent vectors w
(n)
and z
(n)
repeatedly from the equations
 
) ( ) 1 ( n n z w I A
   
R
n n n s ) 1 ( ) 1 ( ) 1 (    
w w z
where s denotes the sign of the first component of w
(n+1)
having the largest absolute value.
The iterations continue until z
(n)
converges. That vector is an eigenvector q
k
corresponding to
the eigenvalue λ
k
.