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

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(35256) (Show Source): You can put this solution on YOUR website!
a) The matrix D is idempotent iff

Since is the matrix where each diagonal entry is squared, and and , this means that EVERY diagonal element will NOT change. So each entry of is equal to its corresponding entry of

So this means that 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 exists and

So if we want A to be idempotent, then we have to show that

So

since

using the associative property

since

because from part a) we proved that

Using the definition that

So since we've shown that , this means that matrix A is idempotent.

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

Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
a) A matrix A is idempotent if . 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 ) 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 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 .
b) , from the condition of (a) above.
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