Jordan Boocks Flashcards
What is a Jordan block J(L,n)
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
What s Jn
Jn = J(0,n)
J(L,n) = LI + Jn
What are ker Jn^k and im Jn^k
Ker Jn^k = span(E1,…Ek) all basis vectors
Im Jn^k = span(E1,…,En-k)
What is M(J(L,n))
M(J(L,n)) = +- characteristic polynomial = (x-L)^n
What are E(J(L,n)(L) and G(J(L,n)(L)
E(J(L,n)(L) = span(E1) (1 dimensional)
G(J(L,n)(L) = F^n
What does any nilpotent operator have
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
What matrix does nilpotent operator phi have
Nilpotent operator phi has matrix Jn1 +o …+o Jnk (1s and 0s on super diagonal and 0s elsewhere)
What matrix does phi have w.r.t to basis of Gphi(Li) and what is size of largest Jordan block
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)
What is sum of sizes of Li Jordan blocks
Sum of sizes of Li Jordan blocks is a.m(Li) n1^i +…+nki ^i = m(i) = a.m(i)
How many Li Jordan blocks are there
Number of Li Jordan blocks is g.m(Li)