Chapter 5.C

Question 1

Diagonal Matrix

Answer 1

A diagonal matrix that is 0 everywhere except possibly along the diagonal

Question 2

Eigenspace, E( λ, T)

Answer 2

Suppose T ∈ L(V) and λ ∈ F.
The eigenspace of T corresponding to λ
denoted E( λ, T) is defined by E( λ, T)=null(T- λ I )

(so E( λ, T) is the set of all eigenvectors of T corresponding to λ along with the 0 vector)

Question 3

Sum of eigenspaces is a direct sum

Answer 3

Suppose V is finite dimensional and T ∈ L(V).
Suppose λ_(1),…,λ_(m) are distinct eigenvalues of T
then E(λ_(1), T)+…+ E( λ_(m), T) is a direct sum
Furthermore dim E(λ_(1), T)+…+ dim E( λ_(m), T) ≤ dim V

Question 4

Diagonalizable

Answer 4

An operator T ∈ L(V) is called diagonalizable if the operator has a diagonal matrix with respect to some basis of V

Question 5

Enough eigenvalues implies diagonalizability

Answer 5

if T ∈ L(V) has dim V distinct eigenvalues, then T is diagonalizable

Question 6

Conditions equivalent to diagonalizability

Answer 6

V is finite dimensional and T ∈ L(V)
let λ_(1),…,λ_(m) denote the distinct eigenvalues of T
then the following are equivalent:
(a) T is diagonalible
(b) v is a basis consisting of eigenvectors of T
(c)There exists 1-dimensional subspaces U_(1),..,U_(n) of V each invariant under T
such that V= U_(1) ⊕ … ⊕U_(n)
(d) V= E( λ_(1), T) ⊕ … ⊕E( λ_(m), T)
(e) dim V = dim E( λ_(1), T) ⊕ … ⊕dim E( λ_(m), T)

Question 7

Direct Sum

Answer 7

Let U, W be subspaces of V.
V = U ⊕ V if
V= U + V and U ∩ W = {0}