Eigenvalues and Eigenvectors

Question 1

What is the definition of an Eigenvalue ?

Answer 1

For a matrix A

an eigenvalue is a value λ such that a non zero vector v ε F is an eigenvector if Av=λv

Question 2

How would you show that an eigenspace is a subspace of F^n ?

Answer 2

Show that the eigenspace is:
Non-empty
closed under addition
closed under scalar multiplication

Question 3

What is the process for finding the eigenspace and eigen vectors of a matrix?

Answer 3

-Form the characteristic polynomial by using:
det(xI-A)=0
-Solve for the required number of λ's
-Solve for the solution space of:
(λI-A)X=0
for every value of λ
This will be the eigenspace
-For each arbitrary parameter in the solution, pick a value and that will be the eigenvector (pick 1)

Question 4

True or False:

Whether or not a system has an eigenvalue depends on field the set is defined over

Answer 4

True

If set over Reals it is possible to get eigenvalues which are complex but cannot be used

Question 5

What is the definition of diagonalisation?

Answer 5

A linear transformation is diagonalisable if V has a basis which consists entirely of eigenvectors

Question 6

How can we know if a linear transformation T is diagonalisable with respect to the dimension of V ?

Answer 6

If dim(V) =n and the linear transformation T has n distinct eigenvalues then T is diagonalisable

Question 7

How would you know if a matrix A and B are similar?

Answer 7

A~B if there exists a matrix P such that

B=P^-1 A P

Question 8

How would you know if a matrix A is diagonalisable ?

Answer 8

A will be diagonalisable if there exists a matrix P such that :
P-1 A P = D 
where 
P = matrix of eigenvectors 
D = diag(eigenvalues)