Definitions Flashcards
adjoint operator
Let V and W be inner product spaces defined over the same field F. The adjoint operator or short adjoint of a linear map T∈L(V,W) is a map T∗:W→V satisfying
⟨Tv,w⟩=⟨v,T∗w⟩
for any v∈V and any w∈W.
algebraic multiplicity
The algebraic multiplicity of an eigenvalue λ of an operator T
is the multiplicity of λ as a root of the characteristic polynomial of T.
canonical homomorphism
Let N be a normal subgroup of a group G. We define a map f:G→G/N by f(a)=aN for any a∈G. Then f is a homomorphism with kernel N.
This homomorphism is called the canonical homomorphism from G onto G/H.
center (of a group)
The center Z(G) of a group G is the set of elements from G that commute with every element from G:
Z(G)={z∈G∣∀a∈G:az=za}
coset
Let G be a group and H a subgroup of G. For a fixed element a∈G, a left coset of H in G is the set
aH:={ah∣h∈H}.
Analogously we define a right coset of H by
Ha:={ha∣h∈H}.
In the following, we use the word coset synonymous for left coset.
cyclic group
A cyclic group is a group that is generated by single element.
factor group
Let G be a group and N a normal subgroup of G. The factor group (or quotient group) G/N is the set of all cosets,
{aN∣a∈G}
equipped with the \alert{multiplication of cosets},
aN⋅bN=(ab)N.
generalized eigenvector
Let k be a positive integer and A be a square matrix with an eigenvalue λ.
Assume (λI−A)k−1x≠0 and (λI−A)kx=0.
Then x is a generalized eigenvector of order k of A.
In the same way, we define a generalized eigenvector of an operator.
generator (of a group)
Let G be a group and X a nonempty subset of G. The subgroup of G generated by X, denoted by ⟨X⟩, is the intersection of all subgroups of G that contain X.
group
A group (G,∘) is a set G together with a binary operation ∘ defined on G such that the following axioms are satisfied:
For all x,y,z∈G, we have (x∘y)∘z=x∘(y∘z) (associativity)-
There exists an element e∈G such that for all x∈G: e∘x=x∘e=e (identity or neutral element).
For each x∈G there exists y∈G such that x∘y=y∘x=e (inverse element).
group product
Let (G,∘) and (H,∗) be groups. The product or direct product of G and H is the group (G×H,⋅) with the group operation
(g1,h1)⋅(g2,h2)=(g1∘g2,h1∗h2).
join (of groups)
Let G be a group and H,K subgroups of G. The join of K and H is the subgroup of G generated by H∪K. It is denoted by ⟨H,K⟩.
linear operator (short: operator)
A linear operator is a linear map (a homomorphism) from one vector space to itself.
Notation: L(V) set of all operators on a vector space V
minimal polynomial
Let V be an F-vector space and T∈L(V). There exists a unique monic polynomial p∈F[t] of smallest degree such that p(T)=0, which is called the minimal polynomial of T.
nilpotent matrix
A square matrix A is nilpotent if there exists a non-negative integer n such that An=0.
nilpotent operator
An operator T∈L(V) is nilpotent if there exists a non-negative integer n
such that Tn=0.
normal matrix (normal operator)
A matrix (an operator) A is normal if A∗A=AA∗.
normal subgroup
Let G be a group and H a subgroup of G. The H is a normal subgroup of G if for all a∈G,
aH=Ha.
skew-symmetric matrix
A matrix A is skew-symmetric if A⊺=−A.
spectral radius
The spectral radius ρ(T) of an operator T is the largest modulus of an
eigenvalue of T:
ρ(T)=max{|λ|:λ∈Spct(T)}
square matrix
A square matrix has the same number of rows and columns.
subgroup
Let G be a group and H⊆G. The set H forms a subgroup of G if H is a group with respect to the operation defined in G.
symmetric matrix
A square matrix A is symmetric if A=A⊺.
Transpose
The transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing another matrix, often denoted by A^T.
[abc] T -> [adg]
[def] [beh]
[ghi] [cfi]
