Page 520 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Appen. C.5 Cholesky Decomposition 507

expressions for a n n X n m atrix:
M 22 — ^99 ^19
22
^33 ~ ^33 ^13 ^23
«44 = - M?4 - M^4 - M^4

«// = 1^ » - E «í;j / = 2, 3 , 4 , . . . , rt

^23 ^ (^23 “ ^ 12^ 13)
^22

^24 “ 7¡ (^24 ~ ^ 12^ 14)
^22
^33
M 3 4 = ~ ( / C 3 4 — W 1 3 W 1 4 — W 2 3 W 2 4 )

/
Wy = 77-(^0 - E W//W;y] / = 2 , 3 , 4 , . . . , n;; = / + 1 ,/ + 2, . . . , «

Inverse of U

The inverse of the triangular m atrix U can be found from the equation:
U U~
(know n) (unknow n inverse) (unit matrix)
U], «12 «13 « 14- Vi2 i’lS Vu '1 0 0 0 ‘
0 ^22 «23 «24 7 " V23 «24 0 1 0 0
V21'' - i ’22
'
0 0 «33 «34 V31 V'32' ^34 0 0 1 0
0 0 0 «44 V41 V42 I'«''^V44 _ 0 0 0 1
Starting the m ultiplication of the two m atrices on the left from the bottom row of
U with the colum ns of u¿j and equating each term to the unit matrix, it w ill be
found that v¿j = 0 for i > \ so that the inverse m atrix is also an upper
j
triangular matrbc. The following sequence of m ultiplication w ill then yield the
following results.
Row 4 X colum ns 1, 2, 3, and 4:
1
l^AA
Í 4 4