B. For the second eigenvector, X2,, from the con- 
dition of ortogonality of eigenvectors XA4 and X 
we can readily write: 
2 K 4,K) @) 
a x? * x ! x(9 * X3 e -0 (27) 
from which it results: 
Kk c 
xe = - oq P X x 2) (28) 
For £2 = 1, the computed value of xe from Eq. 
(28) can then be substituted into Eq. (23b).Hence 
2 1 2 2 k 
x A soi (e ez 9 x Bits J, Coy 
2 2 (29) 
Az= (lyx - xat) n sy - xi^ 
(2,0) 
The system is solved iteratively and for X, 
it results: 
(0) 
27 Gyx =X 
li 
mj 
AK 
a^ cg - xi (30) 
At the second iteration, substituting Eq. (30) in 
to Eqs.(29) it yields: 
(2A 1 2 4, k) (AK 
x. er $x - Tz x - qu, xS ha, 
x5 zi x x2 + x (^^^ (31) 
4 k 
A GE 9 
At the end of the k-th iteration, the solutions 
are: 
qM 1 2 4K) (2k (4,4) 
Zn a Oak? Hx ERLE, xp iy 
e ; x (At) = = 3 iC + x So) 
kl (32) 
hy (Opp = 2,50 2 28) 40% - x 
C. For the third eigenvector XQ , We can readily 
write from the conditions of ~ orthogonality of 
eigenvectors XA, and X),,respectively XJ, and XA: 
xa ra, x (A x 9) + x (4 x À 
x {28x m + te + x 24x 1) - 
For x?) 7 ], it results directly: 
@,s) (2,K), 4 k) (RL C2,K) (Gk), (Kk 
3 Li Xs (3) x ^ KL Xo 
ED meme um cum sums cum eum cam co ue imp ras um m m nm 
(33) 
(4,6) 
0, 14 eB 
= X = 3 
> 
2 XM x^ Ky 3 x Gi) XE x Qni. f k) 
It can be easily derived from Eq.(23c), for i=3, 
that the eigenvalue A3 may be expressed as follows 
3 2 
C 2x +Q0zy xb ( s? 
x 
  
Ja (35) 
The eigenvectors of the covariance matrix 2.j are 
determined with given coefficients of proportio- 
nality. Consequently, all the solutions of the 
system (22) are as follows: 
  
À | xD | xP | x 
Ay [xf c, x9 c, C4 
A2 x Ca C2 x (9 (36) 
A3 C3 x; G^ c 3 yo" 
D. Finally,for checking up the correctness of the 
computation, one may use the equality: 
2 2 2 z 
Mt dar da - Cx Oy+02 = (3p (37) 
For each symmetric matrix Xj, there exists a rota 
tion orthogonal matrix Rj such that R7Z Rj is a 
diagonal matrix where the elements are the corres 
ponding eigenvalues of 23. The columns of R. are 
the normalized eigenvectors Xj} , X},, XA; of 3 j. 
us: 
Top Ya T, = 
R; = vz) Taz T23 VS x, RA, xs (38) 
134 T32 733]j 
  
  
  
  
where: 
[ A] | REPETI 
x, X4/ YX, +X, + X35 
M. x2|- xul KE + XE + X2 ; 
x3 |a, | X yxi e xzeXP 
xA 3. Re 24 7x3 + x3 (2,4) 
SWERE A Kal Ar + XE+ XS ; 099 
| x3 a, RAISERS TRH 
, x) [xy/ Ar + x2+ x3] 6) 
XA, 7 Xp | = | Xo/ YXZ + x3 + x§ 
zy Az | X3/ {+X3+X3|5 
and > 
Xu Fg pking Yl vx Oxy xz 
E SiR; =|Xp2 Xaz Xaa| | x 0 Cyg* 
X45 X25. Raa) | Box Toy ez J 
X414 X42 X 43 N A, 0 0 
HARZ X22. *243] 7.10 45:..9 (40) 
X34 33.2. *23 y 0 0 Az j 
Furthermore, let the orientation of ellipsoid be 
defined by the rotations W, Ÿ and about the X', 
Y' and Z' axes, respectively. 
The three rotation parameters can be obtained by 
identifying the matrices: 
cf cX cV€sX -sÜlsocx sUsx- sYTcocXx 
R4j--cY S4 c«ck -sfsO0SxX swcy + sPcwsx 
sp -cYsw cp cw 
(41) 
from which: 
3.2 r24 
V^ arcsin r,, 3" arctg(- —77); k7 arctg(-. 777) 
r33 r£ 
Cc = cos, s=sin (42) 
In accord with (17), the semiaxes a,b,c of the 
rotated ellipsoid are the square roots of the ei- 
genvalues J, , dg» Az ; respectively: 
a- Gu -VA,; boy - V ce =02'=VAz (43) 
For the probability that the point be situated 
inside or on the ellipsoid defined by (15), where 
a = K yr, b = K 6y!, c = KÜz', the expression is: 
(0 00 O td » + > Ch kàO s BM f/$, lud eB Fa 
Qu FE* 0 C» H