Structure Of Linear Operators

Question 1

When are matrices A,B similar

Answer 1

Matrices A,B are similar if exists non-singular matrix P s.t B = P^-1AP

Question 2

What is a “nice” matrix

Answer 2

“Nice” matrix is 1 with lots of 0s

Question 3

When is a matrix A diagonalisable

Answer 3

Matrix A is diagonalisable iff it has an eigenbasis iff similar to a diagonal matrix with eigenvalues of A as listed according to multiplicity, I.e each Li appears a.m(Li) times

Question 4

When are 2 diagonalisable matrices A,B similar

Answer 4

2 diagonalisable matrices A,B are similar iff have same eigenvalues and multiplicities up to order

Question 5

When is a subspace U phi invariant (phi is a linear operator)

Answer 5

A subspace U is phi invariant iff phu(u) is in U, for all. In U

Question 6

If phi(psi) = psi(phi) (they commute) then what are ker(psi) and im(psi)

Answer 6

If phi and psi commute, then ker(psi) and im(psi) are phi invariant, so are ker(phi) and im(phi)

Question 7

What is direct sum of phi(i)

Answer 7

Direct sum of phi(i) is phi(v) = phi(1(v)) + … + phi(k(k)) for V = V1 +o … +o Vk

Question 8

What is direct sum of square matrices Ai

Answer 8

Direct sum of matrices Ai is
A1 +o … +o Ak = diagonal matrix with diagonal entries A1,…, Ak
This is called block disgonal. See block disgonal example

Question 9

If V = V1 +o … +o Vk and phi = phi1 +o … +o phik then what are properties of phi and Vi

Answer 9

Properties of phi and Bi are:
Phi is linear
Each Vi is phi-invariant
Phi has matrix A1 +o … +o AK w.r.t basis Bi of Vi

Question 10

What is phi/Vi (restriction of phi)

Answer 10

Phi/Vi is :
The map Vi s to V, the function phi is limited to Vi (they behave same range)

Question 11

What is the normal form of a matrix

Answer 11

Normal form of a matrix is when matrix is diagonalised with the E.values on the diagonal listed according to their multiplicities