Hi
Your idea is good
P^-1XPD=DP^-1XP
Let Y be P^-1XP
YD=DY
From part A you know that Y must be of diagonal structure
And X=PYP^-1
Find P and you will get the general form of X
part A(i managed to solve it):
X is a variable of
where
are different rational numbers.
solve XD=DX for X.
solution:
so i get
so for both side to be equal X must be of diagonal structure
every member must be zero except the diagonal
because the lambda values are given as different.
part B(the one that i dont understand):
what is the solution space of XA=AX (use part A)??
i tried
XA=AX
XPDP^-1=PDP^-1X (multiplying by p from the right)
XPDP^-1P=PDP^-1XP
XPD=PDP^-1XP (multiplying by p^-1 from the left)
P^-1XPD=P^-1PDP^-1XP
P^-1XPD=DP^-1XP
another thing i could find is the eigen values of the matrix
i got
and
and
P^-1AP
i can substitute A by X but whats to do next??