User Manual

Section 4: Using Matrix Operations 125
These estimates agree (to within 3 in the ninth significant digit) with the results of the
preceding example, which uses the normal equations. In addition, you can include additional
data and update the parameter estimates. For example, add this data from 1968: CPI = 4.2,
PPI = 2.5 and UR = 3.6.
Keystrokes
Display
1A
1.000000000
Enters row weight for new row.
1¦
2.000000000
Enters x
12,1
.
2.5¦
3.000000000
Enters x
12,2
.
3.6¦
4.000000000
Enters x
12,3
.
4.2¦
1.000000000
Enters y
12
.
B
1.000000000
Updates factorization.
C
3.700256908
|x
13.691900119
Calculates residual sum of
squares.
lC
1.581596327
Displays
)12(
1
b
.
lC
0.373826487
Displays
)12(
2
b
.
lC
0.370971848
Displays
)12(
3
b
.
´•4
0.3710
´U
0.3710
Deactivates User mode.
Eigenvalues of a Symmetric Real Matrix
The eigenvalues of a square matrix A are the roots λj of its characteristic equation
det(A − λI) = 0.
When A is real and symmetric (A = A
T
) its eigenvalues λj are all real and possess orthogonal
eigenvectors q
j
. Then
Aq
j
= λ
j
q
j
and






.1
0
kjif
kjif
k
T
j
qq
The eigenvectors (q
1
, q
2
,…) constitute the columns of an orthogonal matrix Q which satisfies
Q
T
AQ = diag(λ
1
, λ
2
, …)
and
Q
T
= Q
−1
.