Claw-free Graphs Mirrored into Transitive Hereditarily Finite Sets
Sets Finite - University of Toronto
Transcript of Sets Finite - University of Toronto
Finite Sets Finite - dim v. spaces . If
×mn '
If (x ) = fqmalwlineqq.inbos
× Y ""¥' F' "
I:L :p.
""
**one.
Eth- FAIx them.
gotPg.of
"
Compo s . of linear maps off , yis the linen map of Compo of "
Functor
from to VectorSp← categories r
f.d.
Review : V,W v.split ⇒ LIV
,w) 7 A can be described as
a dimwxdimv matrix of elts in # : choose bases
p fnv ,8 for W and write
ma .nn=*e=l::÷÷÷÷÷÷:)Changing fi , 8 will affect the matrix kxk kxn ath
to fir'
⇒ jfa) ,-
-
g. ftp.CAIpplITp '- -
fuk invertible nminvertible
Q =p. p
matrix matrix.
P Q-1
To emphasize the fact that the initial and final kxn
matrices represent the same linear map ,we can define
A,A'
two matrices to be "
equivalent"
when I P,Q invertible
matrices . sit '
A' = PA
Implements edoiumu operations ,invertible#kxk
('
off . - -toI' it
Simplify given kxu matrix A without changing its
equivalence class.
① GE (Row ops) to RRE
( o⇐Ei¥¥E¥¥② GE (Col . ops ) to clear all unknowns (* )
rent
f③ Column switching to bring everything to tight .
I offset diagonal.Thnx:
Every kxn matrixAis equiv .
to
O ←rxr
block matrix #-) r --REA
O O
In terms of bases : for any linear
map A : V→W
F bases f" I sit
.
( e ,. - - ten)
"
( fi,. .. ,f←) .
A- ( e.) = f ,Ale?)= fz
:-
Acer) ='
fr
Ace ;) = o ti > r
rlA7p=ffo)
Recall studied maps X Is X⇒self - hips of a set .
✓
f-. got
a nun,'
l- 2 • 3 q4 represent themap f . in cycle
g y notation (234①-fixed
point cycle
There is a corresponding description for Linear operators"
Linear operators on V"
= L ( V ,V ) .
Key difference : since domain t codomain coincide,
-
we need not choose two bases ;
one basis for V suffices .
i. e.
AE LCV ,V ) has a matrix after singlechoice p = ( e . , -- - , en) basis forV .
Hide -
-
{ It:]matrix .
⇒ have a different equivalence relation on uxu matrices
Def Two nxn matrices A,A'
are similar or conjugatewhen I P invertible man sit
.
-
A'
= PAP"
Motivation if we change f to f'
then
fats ' ' . etat , LIF--
p r> p"
the fact that column ops are forced by the row ops[ means that simplification does not occur by GE . ]⇒ new approach is needed to
"
understand"
the operator: spectral theoryeigenvector analysis
What kind of operators are there?
-
LN ,V) .
ypole =L!! ! ! ingdep .
o) O : v 1-30 .
fr .
basin.
!
1) I : ri- v tr- effs = Igi!
2) ZEE XI : v te Xv
" A. ↳ *← omaha #
a'""tiffs?)
--
,
written in Ad basis1
I i:S", n.
. I µ=ce,p(A)
p ¥> Ae .
K e
. -
-
re, r
lr
( r
-- -
-
While this operator is simple ( easy to understand when written
as above,if we change basia
,won't be simple i
⇒ ⇐÷: Eri;÷µ¥E÷;¥?=":e: same operator !
p'( Ifs PCA ]pp( I ]p , different basis !
E- ( 1¥.
),l-Y://eei.ee) .
In the above example ,e, ,e, play the role of
"fixed pts"
e.
'
,e: net
e., er are examples of Eigenvectors .
the scale factors 1,72 areexamples of
Eigenvalues
4) besides eigenvectors, there is another
type of behaviour :
cydes-e.atto 'detone eigenreet .
⇐% , ez 1-7 e ,
ez 1-1 ez
B -- Ce . . - - ien) bani for ✓
B'-
- Cei,. . .
,en
'
) e! c- V't
--
- i
e : ( en -- I eikjl=loi=j①(ed = 0
:
Pep ( en) -- o
B'islia.indepylet#a±fit-oWTS ai =o fi .
Strategy: show a,=o
,then show 92=0
,~ . -
r
apply LHS to e.
: a,5taze t. .-9= 9
,
RHS to e,
: ocqy = o.
}⇒a
apply LHS to ez
aa,It are
Az
RHS to ee = o,
}arepeat . . .
.D
%% ¥itsapply LHS te e
,
e.'
(ed tea'Ce) = Ito=L
.
RHS =O1=0 # .
Polynomials : x't 1 =/ ( x -7.) (x - Iz)extend 11 / /numbers
real(x - illxti )
once the extension IRC ¢ is done,
we can factorall polynomials
plz) = aot a,Z t 9272 t - - - t an 't"
ai E Cl
= an ( z - H) (z-W -
-a 17 - Xu)
XXe
-
IdF Te Liv,V)
Def: Invariant subspace UE V linear subspace
sit . TCU) E U i.e. Tu E U tf u E U .
✓ °
,
"
i.Tv
,
given TELCV, y
1¥inn
.
Ex⑥ to } E V int .
TreoAreNull
T① Null T
② RangeLT) i.e.
I v EV u-
-Tr
u
U
Tu = TuTu C- Range (Tl .
Deff: an eigenvalue of T is a number TEF
sit.F V nonzero with reNahi
'Tv = .lv/T-aIyIo
"⇒
The set of eigenvalues in tf is called the SPECTRUMnotice this means Spanks) is an Invariant suhspaeeoft
Warning.
.
..
I -:
T OElf is
a valid eigenval .
Any veto in Null (T ) is an eigenvectorwith eigenvalue O
.
Howtofindeigenveotors.su/veAv--Tv
i.e . (A - II ) v = O
2- step procedure-
TO find value I sit.
A - XI
has nullity > oi.e
.A -II not injective
i. e. A - II not surjective .
i.e.rank CA -7 I) c drinkto :÷:÷:÷÷÷::÷: "
( note Null A = So }
Egri A- = ( 28¥ ! ) so o net eigenvalue .)
①A - II '
- f-
{ I, u!) Q: awaken Italian space.
I do GEt
- 21, 2 o
O -T l
27-51714-7702-57114712
02714-71-50217111471-512° °[email protected]:( III)( o →
.
! .ir! ¥o o ingYmthis will have nonzero Null space ⇐ 1=1 or
7=2.⇒ 1=1
,1=2 are
②the a eigenvalues for mi operator . !¥7
choose 1=1 first- plugin explicit value .
Three (Existence of an eigenvalue over F- Q )
PI suppose din V= n choose
any nonzero
vector VE V .
MUST
(v,Tv
,Thr
,Thr
,. - - .
,Tnr) ← BE
UN DEP! ! !
i.e. F ai not allzero sit .
Aorta,Tv t - -
- - t ant"
v I 0
(AOI ta,T t a
,T't - - - t anTn) v = O
#plz) .
view this as a polynomial !with cuffs in field Cl !
= an (T- RI) (T- rat) --
-
- (T- rat ).
v
whoots of plz). in ¢ .
product of ft- r,I) has nonzero
hull space
i.e.
not invertible
⇒ atleast one of the futons is not invertible ,
i. e. (T- ri) has a nonzero null space
⇒ eighth D, =Fi
.
eigenvector is ? ? find it.
-
if we can find a based of
eigenvectorsB : ( e , > . . . , en)
thene
-
- fig ?;%÷ :?)this finding Diagonalof a basis of
Matrix.
eighredfycalled.piag.inalizarin" .
The
V, ,Vz eik with eigenvalues41 Xz and 71¥12 .
Then ( v , ,vz) tin. indep -
①
Pfi suppose a ,v , t am=D
then applyT a.Tv , t qtr , = To-
- O
-②
aid ,r , tazkrz=o
Want to show 9=0 and az=0
to show : take 210 - ②
a. Hair , =o⇒am
-
fo to
a : tube 7,0 -② A
Az ( X ,-7274=0
( 69 ]=A Ae.- e
,
A- ez -_ ez
single eigenvalue 2nd space ofeigenvectors .
If ;) Here .
→- -3
ed .-3 -
←¥i.
←
-
-- l
up µ¥¥Tf1µ - space of states = { up , down } -- S
-- -11C ( s) = { a , - Cup) t a. (down) :
a' laid }down I
e"
basisIasi.
¥9"""ist:÷.
ef II up
In Quantum mechanics,
the light switch
of ( up)-
- 1 has Quantum state = a vector
of ( downy. -1 in ECS)
operator :"
Position"
operator Q-
÷:÷÷:ii÷: !"down ll c l l
le ( i c- Li
att ) -- ⑦ to )classicalstheap