WBEK 4ťf0    SWštíUťR DECOMPOSITION
OF MATRICE
A* - ÁT
A*   - 4r ^
A i'S MpHvXj.aAts   (gfTft/mECu'o)    ■<■/-     A-A .
ítúMjAu :     A' /Ut: "3
4-i hui
* _ f2-C    4+č   ~ ť
-3    4-2ľ é-Sí'
A
Lem*<l     &6 A M, * /*hsUw^ X^. T/».    AA* & W**^ ílK) ^
/rastúc*)   a Uf^^^ <zku ^0.
-z-
Jwt   A* A  = <*>eo A
/f
y-e^^oA, then A*A* X fe leu, A*A . Xfe £^,4*4 , they) A*A*~0
= A*Ú~Ú
i
tf AY* =0 A/
/•e. X6 ker A .
Tftg^g^ °h £(y>$Hl4.ir decomposition Let A e WaCMmj  W).   Then there, are.    uni tatra ( orthoflonaC) matrices
2M<2<1 that
a -		
		
	5, 0	0
		
	I 0	1 0
fi,(..(ir   Are ß1t f>ír. f\, *£ positive, eigenvalues   of   the   matňy.     A*A .
(D
Troof    qľtes   the wac<   hoix/ to ftv,^ this
Consider y : —
Then   <{*<}>:  (k* —7IK*" t <f*pM~A+Ay
/S 4. Ceif adjoint    opetečoiŕ   U/ifch noHHôQitiľc^
•ťiqevi values. .   There t'c <u ortbehamxzC hosts     ot ^ Lki, ... ut, Hu,,.- o in ^omeoi   fw»i eigenvectors   mth efyehva/t«^
Vectors    f ĹHi }J0&4te9 / cf («*.) t.. <p(ur) at&
j>at   vc -   igM ^/ér
a h fit complete- is<eator    v1t ^ ..      ť" to
1l'
-h -
c f- a* a . The orto^ommnC bims jlomasŕ        eigenvalues,      flí'ves  í he.
faalrit    Q   ~ ( «i     t... **) .
The   ím&f&& vectors    Anu Au^,.. 4^
4hs   £o>/   complete   th*** ľ»é0 <Oirčo-
Tseubo/n/i/fRsf A/AT PIX
A & hal^ UK)    is   A'1   čuch t fat?
A A' 6mit i*ia£K>)
Where* ^      ( $i ^ o
0 \s
to
Then        4-1 =(>sc?*\-i =
- (Q+y š-> V" =
(/š/ha this,   ohse'ir^dtt'oa    u/e can  feuern-\ the   atefiWdri'O*    +r- t'*r*lre& l-afai
pg-f     y.   A & W**^ ü£) i ůteh the. toceiAdoinverse.    A^'        A Cs
'"f      a  . -? fD   °\ 6?*
*Vroferhl&£ & f- )f> üeudoin verges, (f)    l£   A   is    ľnverbľbte c éh&m
A C'4)  - A"1
OA orbo^omt   projection ><
©    (ä    m - A .
^ a a    a  -a. 1
®
a('1) -  (a'a)1'^ a*
Co\ňôemeVic&       Of    A1-A    is infertil*/^
éheh a'">    = (a*Af. a*
and  we   &ih   confute    aĺ''] mä»°"£
'firín&ittfA-noy ^zomrto**
0P     U/V.    £F$UA HO A/S Leh    A &  MA-čirt')*   A-íi^. CoHStMctr
b       Her, kefc^
ó f     vahté (Ali) > rant CA) the   Qüsb&*t ho   eolubt'c* . t"
Otze   we   wo**   to  r/W x^*"" thd<r    M*- b l|   r s   mi *i tu«.!. tk^toea**,
TH w     ?%<s   function   II 4 /es minimum    in     X - b
Thí& šycbewi has. ýto šolu-trev . ttevevtUe/^ Vc/e aire   IqoíCia^     fo\r *  Qucln irfalij
h-'ii \
-^^Tdecomposition &r matrice^
frevf Aék °^
can   be vm'He»
se mede fi m be U * orto»™»*
£>\r   lAwibairifr.     Oj-   A  ľs   ľr>vetrtibl& (
i4 ^ a uzÁ^y