Chapter 8

Question 1

generalized eigenvector

Answer 1

suppose T in L(V) and λ is an eigenvalue of T.

A vector v in V is called a generalized eigenvector of T corresponding to λ
if v ≠ 0
and ((T - λId)^j)*v = 0 for some positive integer j

Question 2

generalized eigenspace (G(λ, T))

Answer 2

Suppose T in L(V) and λ in F.

The generalized eigenspace of T corresponding to λ, denoted G(λ, T), is defined to be the set of all generalized eigenvectors of T corresponding to λ,

along with the 0 vector

Question 3

nilpotent

Answer 3

an operator where some power of it equals 0

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Question 4

description of operators on complex vector spaces

Answer 4

Suppose V is a complex vector space and T in L(V).
Let λ1,…,λm be the distinct eigenvalues of T.

Then:

(a) V = G(λ1, T) (+) … (+) G(λm, T)
(b) each G(λj, T) is invariant under T
(c) each (T - λjId) restricted to G(λj, T) is nilpotent

Question 5

Jordan basis

Answer 5

Suppose T in L(V).
A basis of V is called a Jordan basis for T if w.r.t. this basis T has a block diagonal matrix ((A1,…,0),…,(0,…,Ap)),

where each Aj is an upper-triangular matrix of the form Aj = ((λj, 1, …, 0),…,(…,1),(0,…,λj))
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Question 6

Jordan Form

Answer 6

Suppose V is a complex vector space.

If T in L(V),
Then there is a basis of V that is a Jordan basis for T

Question 7

description of generalized eigenspaces

Answer 7

Suppose T in L(V) and λ in F.

Then G(λ, T) = ker(T - λId)^(dim V)

Question 8

matrix of a nilpotent operator (useful result)

Answer 8

suppose N is a nilpotent operator on V.

Then there is a basis of V w.r.t. which the matrix of N has the form ((0,…,*), …, (0,…,0)); here all entries on and below the diagonal are 0’s

Question 9

a basis of generalized eigenvectors (useful result)

Answer 9

Suppose V is a complex vector space and T in L(V).

Then there is a basis of V consisting of generalized eigenvectors of T

Question 10

sum of the multiplicities equals dim V

Answer 10

Suppose V is a complex vector space and T in L(V).

Then the sum of the multiplicities of all the eigenvalues of T equals dim V

Question 11

Multiplicity of an eigenvalue

Answer 11

The dimension of the corresponding generalized eigenspace