⑪
LV factorization
1/ Eri I
- 2 o u
A = G
E ,
A
[] EstI=
EE ,
A
= E
E - I
=
F -9
y
*x =
1b =
(2)
I
LUX =
UX =
2
-
Le =
b save for 1 and then
UX =
1 for X .
②
Le =
(5) = =
()
x = 2
, y
= 0
,
z =
10 L(z)
=
(i) :] =
1)
I (i).
X =
[
Does LV factorization always work !
Example :
-
/100 I
2 12
A = -
32/
Swap
-
rows :
P() (2) =
D =
[
it : 31 ,
2
, .., nb > 51, 2
..., my
It is invertible permutation ③
(123)
23/
I =
( =
/- I
C C
PA =
Perform LV factorization for
this metrix
PA = Lu
=] v
=
AX =
b =>
PAX =
Pb
Lux =
PE
-
PA permutes rows
AP 1)
columns .
[h/[56] =
[I
④
(AT)ij =
Aji
1: : =
12
=A .
(A + B)T =
AT+ BT
(AB)T = BTAT
(B)
Tij =
B) ji
=
z Aj Bri
k
(T)kj /BT)ik
=
(BT)in
(T)kj
=
(BTAT)
ij
(AT) =
(A-Yi
⑤
AA" =
I = (A) =
I
T
=
CA") AT =
I
↑A-)
+
is left inves of AT
AA =
I = (A A)T = I o
A
+
A )T =
I
=>
At T is
right inverse
or (AT) =
<A")
T
.
(Ax)T =
XT AT
↑
AX linear combination of columns. ofK
* TAT linear combinations of
the rows of AT
.
Dot producti
XTAy =
(A)T]
#T
A =
symmetric
A myn
> > At nxm
⑥
of IR
-cubspaces
V subspace of IR" .
u ,
veV = A + 1EV
GEV ,
J Scalar = JEV .
Us ,
Unc-- > AktV C . -
, 2 Scalers
K
Ester
Amxn
·
CLA) column space is a subspace
of IRM
· NCA) =
SEIR" : Ax =
03 ,
NCA) is a subspace of IR".
*,, Xz N(A)
A(X, 12)
#+
=>
+
&
= =
.
O
-
G
↓ ENIA) a scalar .
↑ (1)
d Es
=
⑦
I e
12
A =
I
1 -
1 -
12
/2 J I ·
01 2 ( C 12
, 2 C
-> - O I
O 12 C C O C
C C 00
0 -
1 -
2 C
Ro
() (
2z 0x + z + zw= 0
y + =
z(-2) w() +
1/
special solutions .
besis for
they are
a
me
hull space.
⑧
Basis V subspace
V
-
1 . .... Up is a beas if
Ev were ex <, --
E with
.
,
k
= Es
j = 1
,
i . e
. , .... Uk
Spen .
V
1. -- > As are .
knearly independent
The special solutions of Ax =*
form a basis for NCA)
.
)N(Rd) =
not
solution space does change
under now reduction
.
CCA)
- ((Ro)
I
-
I
11 2
I
O
10 12
I
2 120
A =
C ↓ zu R
j =
O 00 O
1 32 O oo G
L -
1 -
12
Ro :
In ⑤
1 2(g)-1:1 /:+ =
Ro() 1:↑ =
A
(=2) v, +
24 1 g
-
=
=
24
+
-
A-1, o ↓
2
&, &2 , As &4
&, 82 , &3 =
&, +
242
&4
= 29 ,
A =
[a ,
a) Id ? If
C R
⑩
It will turn out that in the
CR factorization R will be
the reduced echelon form
The number of pirchs =
# of
knearly indep columns that
Spen CCA) .
# of free veviables =
# of
special solutions
. # of
columns
.
↓
#of Hirots + # of special sol .
=
dimCCA) + chimNCA) =
.
n
Fundamental tim of him .
algebre
dimCCA) = renk of A
i 114
(1 , 92 , 13 &4)
,
I
A, diz 93 Alt
an
031
C+
d2z
Az AAs
Ays 924
a+ 4
I
