SOLUTION: Could you please help me with this problem: Let D be an n x n diagonal matrix whose diagonal entries are either 0 or 1. a) show that D is idempotent b) Show that if X is a n

Algebra ->  Algebra  -> Matrices-and-determiminant -> SOLUTION: Could you please help me with this problem: Let D be an n x n diagonal matrix whose diagonal entries are either 0 or 1. a) show that D is idempotent b) Show that if X is a n      Log On

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Question 376125: Could you please help me with this problem:
Let D be an n x n diagonal matrix whose diagonal entries are either 0 or 1.
a) show that D is idempotent
b) Show that if X is a nonsingular matrix and A = XDX^(-1), then A is idempotent.
Found 2 solutions by jim_thompson5910, robertb:
Answer by jim_thompson5910(28717) About Me  (Show Source):
You can put this solution on YOUR website!
a) The matrix D is idempotent iff D%5E2=D


Since D%5E2 is the matrix where each diagonal entry is squared, and 0%5E2=0 and 1%5E2=1, this means that EVERY diagonal element will NOT change. So each entry of D%5E2 is equal to its corresponding entry of D


So this means that D%5E2=D and that the matrix D is idempotent.


Unfortunately I can't type out generalized matrices here, but hopefully you can see it.


b)

Let's assume that X is nonsingular (ie X is invertible). So this means that X%5E%28-1%29 exists and X%2AX%5E%28-1%29=I


So if we want A to be idempotent, then we have to show that A%5E2=A


So A%5E2=%28XDX%5E%28-1%29%29%5E2


A%5E2=%28XDX%5E%28-1%29%29%28XDX%5E%28-1%29%29 since X%5E2=X%2AX


A%5E2=XD%28X%5E%28-1%29X%29DX%5E%28-1%29 using the associative property


A%5E2=XD%28I%29DX%5E%28-1%29 since X%2AX%5E%28-1%29=I


A%5E2=XD%2ADX%5E%28-1%29


A%5E2=XD%5E2X%5E%28-1%29


A%5E2=XDX%5E%28-1%29 because from part a) we proved that D%5E2=D


A%5E2=A Using the definition that A=XDX%5E%28-1%29


So since we've shown that A%5E2=A, this means that matrix A is idempotent.


Hopefully this is clear. If not, let me know.



Answer by robertb(4012) About Me  (Show Source):
You can put this solution on YOUR website!
a) A matrix A is idempotent if A%5E2+=+A. Let D = [d(k,j)] be a diagonal nxn matrix Suppose the diagonal entry d(k,k) = 0. Then the dot product of the kth row of D and the kth column of D (when we're determining D%5E2) is also 0 (because the rest of the entries in that row and column are zeroes).
If d(k,k) = 1, then the dot product of the kth row of D and the kth column of D is also 1, (again because the rest of the entries in that row and column are zeroes). Thus D%5E2 and D have the same diagonal elements. But the product of any two compatible diagonal matrices is also a diagonal matrix, and thus all the off-diagonal entries are also zero. Therefore D%5E2+=+D.
b) , from the condition of (a) above.