THEOREMS Flashcards
v(^)j
- omitting vj from list
- ## if list was linearly indep then so is ommited list
- THEOREM 2.1.9
- COR. 2.1.10
- if β=u1,…,un basis for V and γ=v1,…,vr is linearly independent, then r<=n
if r=n then γ basis for V - if v1,…,vn and w1,…,wr are bases of V then r=n
- COR. 2.2.8
- THEOREM 2.2.9
- Let V n-dim and β=v1,…,vn∈V
- if β spans V, then it is a basis
- if β linearly independent then it is basis
- U subspace of n-dim V. then U finite dim. with dimU<=n
dimU=n ⟺U=V
COR. 2.5.3
V,W finite dim. with dimV=dimW and T∈L(V,W)
then KerT={0} ⟺ ranT=W
THEOREM 5.2.5
V finite dim. inner product space
β=u1,…,un orthonormal basis
v∈V
then v= Σ<v,ui>ui
||v||^2 = Σ|<v,ui>|^2
and basis representation of v w.r.t. β is [v]_β = [<v,u1> … <v,un>]^T
COR. 5.3.12
COR 5.3.13
- every finite dim. inner product space has an orthonormal basis
- every orthonormal system in a finite dim. inner product space can be extended to an orthonormal basis
THEOREM 7.1.8
if U finite dim. subsoace of innerproduct space, then (U^(⊥))^(⊥)=U
THEOREM 7.3.9
Let X, Y∈Mn×r have orthonormal columns
Then col X = col Y ⟺ ∃U ∈ Mr unitary: X = YU.
THEOREM 7.3.10
Let U be a finite-dimensional subspace of V
- P_{0} = 0 and P_V = I
- ran P_U = U
- ker P_U = U^(⊥)
- v − P_U(v) ∈ U^(⊥), ∀v ∈ V
- ||P_U (v)|| ≤ ||v||, ∀v ∈ V
(equality ⟺ v ∈ U)
THEOREM 7.3.11
let V be finite dim., U subspace of V
1. (P_(u)^(⊥)) = I - (P_u)
2. (P_u)(P_(u)^(⊥))=(P_(u)^(⊥))(P_u)=0
3. (P_(u))^(2) = (P_u)
4. (P_u) = (P_(u))^*
THEOREM 7.3.14
let V be finite dim., and P∈L(V)
then
P orthogonal projection onto ranP
⟺
P self adjoint and idempotent
THEOREM 13.1.2
A∈Mn(F) positive semi-definite
⟺
A hermitian and all eigenvalues non-negative
⟺
∃B∈Mn(F) : A=BB
⟺
for some m>0, ∃B∈Mmxn(F): A=BB
THEOREM 13.1.8
A∈Mn(F) positive definite
⟺
A hermitian and all eigenvalues positive
⟺
A is positive semi-definite and invertible
⟺
A is invertible and A^(-1) is positive definite
⟺
∃B∈Mn(F) invertible : A=BB
⟺
for some m>=n, ∃B∈Mmxn(F): rankB=n and A=BB
THEOREM 13.1.9
- let A∈Mn(F) positive semi-definite
then
1. trA>=0 with strict equality ⟺ A≠0
2. detA>=0 with strict equality ⟺ A positive definite
THEOREM 13.1.10
suppose B∈Mmxn(F) and let A=BB
then
1. nullA=nullB
2. rankA=rankB
3. A=0 ⟺ B=0
4. colA = colB
5. if B normal then colA=colB
cor. 13.1.12
let A∈Mn be positive semidefinite and partitioned as
A=[[A11 A12][A12*][A22]]
then A11 is positive semidefinite and
colA12 ⊆ col A11
consequently
rankA12<=rankA11
if A is positive definite then A11 is positive definite
cor 13.1.18
let A∈Mn be positive definite and let x∈Cn
then <Ax,x>=0 ⟺ Ax=0
theorem 13.1.20
let A,B∈Mn(F) and a,b∈R
- if A,B hermitian then aA+bB hermitian
- if a,b nonnegative and A,B are positive semidefinite, then aA+bB positive semidefinite
- if a,b positive, A,B positive semidefinite, and at least one of A,B are positive definite then aA+bB positive definite
theorem 13.1.23
let A,S∈Mn, B∈Mm, suppose A,B positive semidefinite
- A+B positive semidefinite,
(positive definite ⟺ A,B both positive definite) - S*AS positive semidefinite,
(positive definite ⟺ A positive definite and S invertible)
theorem 13.1.24
let A=[aij]∈Mn hermitian, suppose has nonnegative diagonal entries
- if A diagonally dominant, then positive semidefinite
- if A diagonally dominant and invertible, then positive definite
- A strictly diagonally dominant, then positive definite
- A diagonally dominant, has no zero entries, and |akk||>R’k(A) for at least one k∈{1,…,n} then A positive definite
THEOREM 14.1.3
let A∈Mmxn(F) with rankA=r and q=min{m,n}
- δ1^2,…, δr^2 are positive eigenvalues of A
- Σδi^2 = trAA = trAA = (||A||F)^2
- A, A*, A^T and A() have the same singular values
- the singular values of cA are |c|δ1,…,|c|δq
