Definitions

Question 1

adjoint operator

Answer 1

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.

Question 2

algebraic multiplicity

Answer 2

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.

Question 3

canonical homomorphism

Answer 3

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.

Question 4

center (of a group)

Answer 4

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}

Question 5

coset

Answer 5

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.

Question 6

cyclic group

Answer 6

A cyclic group is a group that is generated by single element.

Question 7

factor group

Answer 7

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.

Question 8

generalized eigenvector

Answer 8

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.

Question 9

generator (of a group)

Answer 9

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.

Question 10

group

Answer 10

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).

Question 11

group product

Answer 11

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).

Question 12

join (of groups)

Answer 12

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⟩.

Question 13

linear operator (short: operator)

Answer 13

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

Question 14

minimal polynomial

Answer 14

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.

Question 15

nilpotent matrix

Answer 15

A square matrix A is nilpotent if there exists a non-negative integer n such that An=0.

Question 16

nilpotent operator

Answer 16

An operator T∈L(V) is nilpotent if there exists a non-negative integer n
such that Tn=0.

Question 17

normal matrix (normal operator)

Answer 17

A matrix (an operator) A is normal if A∗A=AA∗.

Question 18

normal subgroup

Answer 18

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.

Question 19

skew-symmetric matrix

Answer 19

A matrix A is skew-symmetric if A⊺=−A.

Question 20

spectral radius

Answer 20

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)}

Question 21

square matrix

Answer 21

A square matrix has the same number of rows and columns.

Question 22

subgroup

Answer 22

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.

Question 23

symmetric matrix

Answer 23

A square matrix A is symmetric if A=A⊺.

Question 24

Transpose

Answer 24

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]

Question 25

Orthogonal matrix

Answer 25

In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.

One way to express this is
Q^T Q = Q Q^T = I,
where Q^T is the transpose of Q and I is the identity matrix.

This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse:
Q^T =Q^-1,
where Q^−1 is the inverse of Q.