110.460 Random Matrices Homework Answer KEY 1
3.A box contains three white and two black balls. Two balls are to be drawn from
the box. Describe a two-dimensional sample space and assign probability to the sample
points when (a)the balls are also numbered
110.460 Random Matrices Homework Answer KEY 4
1.
pf: H = H , H = H R = JH T J 1 , here J =
0N 1N
1N 0N
.
Now if x1 is the normalized eigenvector such that Hx1 = 1 x1 , we will have H(Jx ) =
1
JH T x = J(H x1 ) = J(Hx1 ) = J(1 x1 ) = 1 Jx . So Jx is a also
110.460 Random Matrices Homework Answer KEY 6
1.Prove COEN U (N )/O(N ).
proof:COEN = cfw_S U (N ) : S = S T
(a)If is an eigenvalue of S, and x is an eigenvector of , i.e. Sx = x, then
1
Sx = (S T ) x) = (S x) = (S 1 x) = ( x) = ( x) = x . This implies S
110.460 Random Matrices Homework Answer KEY 2
1.
pf:
(a)For any a, b C and , V , we have A (a+b) = CAC(a+b) = CA(a (C)+
b (C) = C(a (AC) + b (AC) = aC(AC) + bC(AC) = aCAC() + bCAC() =
aA () + bA ().
(b)Let A be a real operator and be a real vector. By the
110.460 Random Matrices Homework Answer KEY 7
1.W R W = I J T W T JW = I JJ T W T JW = JI W T JW = J.
2.Let S Sp (N ). From S R S = I, we know (dS)R S + S R dS = 0, i.e. (dS)R S =
(S R dS). And (S R dS)R = J T (S R dS)T J = J T (dS)T (J T S T J)T J = (dS)
110.460 Random Matrices Homework Answer KEY 3
1.
pf: A matrix H in GU E satises (1)P (H)dH = P (H )dH if H = U 1 HU where U is
(0)
(1)
(0)
(1)
any unitary matrix. (2) P (H) = ij Pij (Hij ) i<j Pij (Hij ); here Hij , Hij are real
and image part of Hij .
Th