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Appendix I / Utilities / Matrices
_ _
22.9.19 Matrix Symmetry
M_SYMMETRIC takes a near-symmetric matrix [P(n,n)], and returns it with
its off-diagonal elements symmetric using the method determined by (k).
M_SYMMETRY
22.9-6
( [ P ] , n , k )
Equation 22.9-36
For the case where the lower triangular matrix is loaded into the upper
triangular matrix,
( k = 0 ) ∧ ( i ∈ [ 1(
1 ) n − 1]
) ∧ ( j ∈ [ 1 + 1(
1 ) m ] )
p
( i,
j ) : = p(
j,
i )
Equation 22.9-37
For the case where the average value of the upper and lower off-diagonal
elements are loaded into the respective triangular matrices,
( k ≠ 0 ) ∧ ( i ∈ [ 1(
1 ) n −1
] ) ∧ ( j ∈ [ 1 + 1(
1 ) m ] )
⎧
⎪
⎨
⎪
⎩
22.9.20 Matrix Positive Define Testing
p
( i,
j ) : = 0.
5 ⋅ ( p ( i,
j ) + p ( j,
i ) )
p
⎫
⎪
⎬
( j,
i ) : = p ( i,
j ) ⎪
⎭
⇒
⇒
Equation 22.9-38
M_POSDEF takes covariance matrix [C(n,n)], and returns a value > 0 when
it is ill-conditioned, otherwise zero. A non-zero value indicates the type of
ill-conditioning. If k > 0 [C] is returned after forcing positive-definiteness.
y : = M_POSDEF
( [ C ] , n , k )
First test the main diagonal for zero, or negative terms,
( i ∈ [ 1 ( 1 ) n ] ) ∧ c ( i , i )
Equation 22.9-39
−8
( < 10 ) ⇒ y : = 2
The 2 nd stage is to apply the off-diagonal correlation test,
Equation 22.9-40