Diagonalisation

Question 1

When is a linear operator phi diagonalisable

Answer 1

Linear operator phi diagonalisable if :

E basis alpha of V consisting of eigenvectors of phi (alpha is called eigenbasis of V)

Question 2

When is a matrix A diagonalisable

Answer 2

Matrix A is diagonalisable when:
E matrix P s.t P^-1AP is diagonal (A is diagonalised by P)
This is equivalent to previous linear operator flashcards

Question 3

What is formula for diagonal matrix D^k

Answer 3

Formula for diagonal matrix D^k is:
All entries raised to kth power
So A^k = PD^kP^-1

Question 4

When are V1,…,Vn eigenvectors of phi L.I

Answer 4

V1,…Vn are L.I when they have distinct eigenvalues L1,…Lm

Question 5

What is algebraic multiplicity

Answer 5

Algebraic multiplicity is:

Largest power of L-t that divides characteristic polynomial

Question 6

What is geometric multiplicity of L

Answer 6

Geometric multiplicity of L is

Dim(E(phi)) where E(phi) is L-eigenspace of phi

Question 7

What is relationship between a.m(L) and g.m(L)

Answer 7

Relationship between a.m(L) and g.m(L) is:

A.m(L) >= g.m(L)

Question 8

When is a linear operator phi diagonalisable

Answer 8

Linear operator phi is diagonalisable when:

Characteristic polynomial is a product of linear factors and a.m(L) = g.m(L) for all eigenvalues L