T in L(V), λ’s distinct e-values, s distinct e-values,
t.f.a.e. (1) T is diagonalizable
( 2)V has basis consisting of
e-values (3)Exists 1-d subspaces
Ui, T-invariant, V =@Ui
(4) V=@N(T-λiI) (5) dimV =
ΣdimN(T-λdimN(T-λiI) (6) pT has no
repeated roots
Cayley Hamilton Thm: XT(T) =0
in L(V) for any F, V=Z(v,T) =
span(v,Tv,T2v,…) pT=XT
XT=a +bx+cx2+x3
Defn: pT is monic poly of least
deg s.t.pT(T)=0
Defn: XT= det(xI-[T]β) =
Defn: e(λ) = dim(gen e-space of
λ) = dimN(T-λI)n
Structure Thm for Complex
Operators: T in L(V), λ’s distinct e-values, s distinct
e-values, Ui gen e-space for λi.
Then
(1) V=@Ui (2)Ui T-invariant (3)
Each Si = (T-λiI)|Ui is nilpotent
(4)T=D+S (5)DS=SD
q nonzero, q(T) =0. Then p∙q nonzero, q(T) =0. Then pT|q
p∙q nonzero, q(T) =0. Then pT is unique
deg p∙q nonzero, q(T) =0. Then pT and XT is <=n2 in all F
deg(p∙q nonzero, q(T) =0. Then pT) <=n and deg(XT) =n in
C.
↑Δ, then λ appears on the ∙q nonzero, q(T) =0. Then p
diagonal e(λ) times
Structure Thm for Nilpotent
Operators: F , S nilpotent in
L(V). Then there exists a
collection of vectors vi nonzero
s.t. V=@Z(vi,S). Also <Sm(v1)(v1), … ,
Sm(vk)(vk)> is a basis for N(S).
If S is nilpotent, then exists a ∙q nonzero, q(T) =0. Then p
basis such that [S]β is strictly
upper triangular
Complex Spectral Thm: T
normal (TT*=T*T) V has an
o.n. basis of eigenvectors of T.
C, T has at least one ∙q nonzero, q(T) =0. Then p
eigenvalue
C, T=T* ∙q nonzero, q(T) =0. Then p <T(v),v> in R for all
v in V
C, T=0 ∙q nonzero, q(T) =0. Then p <T(v),v> = 0 for all v
in V
C, ↑Δ for some o.n. basis of V∙q nonzero, q(T) =0. Then p
Real Spectral Thm: T=T* V
has an o.n. basis of eigenvectors
of T.
R, T=T* => T has at least 1 e-∙q nonzero, q(T) =0. Then p
value
λ e-value of T ∙q nonzero, q(T) =0. Then p is an e-
value of T*
e-vectors of distinct e-values ∙q nonzero, q(T) =0. Then p
are L.I.
T Normal
Tv = λv ∙q nonzero, q(T) =0. Then p T*v = v
eigenvectors w/ distinct ∙q nonzero, q(T) =0. Then p
eigenvalues are orthogonal
||Tv|| = ||T*v|| for all v in V∙q nonzero, q(T) =0. Then p
Properties of Adjoint
(1) (S+T)*=S*+T* (2)(aT)*= T*
(3) (T*)*=T (4)(ST)*=T*S*
N(T*) =R(T)perp and R(T*) = ∙q nonzero, q(T) =0. Then p
N(T)perp
[T*]∙q nonzero, q(T) =0. Then pβ = βt
adjoint is unique∙q nonzero, q(T) =0. Then p
Properties of Projection
(1) R(P)=U, N(P)=Uperp (2) P2=P
(3)v-P(v) is in Uperp (4) ||Pv||
<= ||Pv|| + ||w|| = ||Pv+w||
= ||v|| (5) Pv is closest vector
to V that lies in U
Change of basis
[T]γ = [Id]βγ[T]β[Id]γβ = P-1[T]βP
Similar matrices have the ∙q nonzero, q(T) =0. Then p
same XT
in R∙q nonzero, q(T) =0. Then pn and Cn, <v,w> = vt
Every finite dim inner product ∙q nonzero, q(T) =0. Then p
space has an orthonormal basis.
Every o.n. list can be extended∙q nonzero, q(T) =0. Then p
to and o.n. basis of V
All eigenvalues of a self adjoint∙q nonzero, q(T) =0. Then p
matrix are real
Let β=(ei) be an o.n. list of
vectors. Then v in span(β) => v =
ΣdimN(T-λ<v,ei>ei
and β is independent.
Gram-Schmidt Procedure:
If (vi) is a independent list then
there exists an orthonormal list
(ei) s.t. span(vi) = span(ei).
e∙q nonzero, q(T) =0. Then p1=v1
∙q nonzero, q(T) =0. Then p
Dual and Annihilator
V≈V*≈V**, for finite ∙q nonzero, q(T) =0. Then p
dimensional V
σ = eval∙q nonzero, q(T) =0. Then pv is the natural
isomorphism from V to
V**(finite dim)
σ is even injective in the ∙q nonzero, q(T) =0. Then p
infinite dim
σ(f) = eval∙q nonzero, q(T) =0. Then pv(f) = f(v)
T*(f) = fT ∙q nonzero, q(T) =0. Then p
T:V->W, T*:W*->V*, f:W->F∙q nonzero, q(T) =0. Then p
N(T*) = R(T)⁰∙q nonzero, q(T) =0. Then p
dim(V) = dim(U) + dim(U⁰)∙q nonzero, q(T) =0. Then p
[T*]∙q nonzero, q(T) =0. Then pβγ = [T]γβt
Let β=(v∙q nonzero, q(T) =0. Then pi) be a basis for V.
Then β*=(fi) a basis of V* w/
fi(vi) = δij
