Jordan Boocks

Question 1

What is a Jordan block J(L,n)

Answer 1

Jordan block J(L,n) is matrix with L’s on diagonal, 1’s on super diagonal (diagonal above leading diagonal) and 0’s elsewhere

Question 2

What s Jn

Answer 2

Jn = J(0,n)
J(L,n) = LI + Jn

Question 3

What are ker Jn^k and im Jn^k

Answer 3

Ker Jn^k = span(E1,…Ek) all basis vectors
Im Jn^k = span(E1,…,En-k)

Question 4

What is M(J(L,n))

Answer 4

M(J(L,n)) = +- characteristic polynomial = (x-L)^n

Question 5

What are E(J(L,n)(L) and G(J(L,n)(L)

Answer 5

E(J(L,n)(L) = span(E1) (1 dimensional)
G(J(L,n)(L) = F^n

Question 6

What does any nilpotent operator have

Answer 6

Any nilpotent operator has a basis w.r.t which phi has matrix Jn1 +o … +o Jnk (Jordan blocks) where n1+…+ nk = Dim V no of Jordan bLocks

Question 7

What matrix does nilpotent operator phi have

Answer 7

Nilpotent operator phi has matrix Jn1 +o …+o Jnk (1s and 0s on super diagonal and 0s elsewhere)

Question 8

What matrix does phi have w.r.t to basis of Gphi(Li) and what is size of largest Jordan block

Answer 8

Phihas matrix J(Li,n1^i) +o …+o J(L,nki^i) where n is size of matrix
Size of largest Jordan block is Si where M(phi) = PI i = 1 to k (x - Li)^si 1<= si <= mi = a.m(Li)
Ki = dim E(Li) = g.m(Li)

Question 9

What is sum of sizes of Li Jordan blocks

Answer 9

Sum of sizes of Li Jordan blocks is a.m(Li) n1^i +…+nki ^i = m(i) = a.m(i)

Question 10

How many Li Jordan blocks are there

Answer 10

Number of Li Jordan blocks is g.m(Li)