Jordan Blocks 2

Question 1

What do all nilpotent operators on finite dim V

Answer 1

All nilpotent operators on finite dim V have a matrix Jn1 +o … +o Jnk, (phi^n = 0)

Question 2

What is M(phi)

Answer 2

M(phi) = x^s where s = max (n1,…,nk) (of Jordan blocks)

Question 3

What is a Jordan basis

Answer 3

Jordan basis is a basis of V w.r.t which linear operator phi has matrix a direct sum of jordan blocks

Question 4

What is jordan normal form of phi

Answer 4

Jorda normal form of phi is sum of Jordan blocks

Question 5

What is M(phi) if phi is a linear operator

Answer 5

M(phi) if phi is a linear operator is
PI I=1 to k (x - Li)^si where Si is the largest Jordan block of phi with eigenvalue L

Question 6

When is phi diagonalisable

Answer 6

Phi is diagonalisable iff M(phi) = PI I = 1 to k (x - Li) all Si = 1

Question 7

When is any matrix A similar

Answer 7

Any matrix A is similar to a direct sum of Jordan blocks, i.e Exists P s,t P^-1AP = A1 +o … +o Ai (jordan normal) where each Ai is Jordan block,

Question 8

When are matrices A and B similar

Answer 8

Matrices A and B are similar when they have same JNF (jordan normal form) up to same order