Question 376125
a) The matrix D is idempotent iff {{{D^2=D}}}



Since {{{D^2}}} is the matrix where each diagonal entry is squared, and {{{0^2=0}}} and {{{1^2=1}}}, this means that EVERY diagonal element will NOT change. So each entry of {{{D^2}}} is equal to its corresponding entry of {{{D}}}



So this means that {{{D^2=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^(-1)}}} exists and {{{X*X^(-1)=I}}}



So if we want A to be idempotent, then we have to show that {{{A^2=A}}}



So {{{A^2=(XDX^(-1))^2}}}



{{{A^2=(XDX^(-1))(XDX^(-1))}}} since {{{X^2=X*X}}}



{{{A^2=XD(X^(-1)X)DX^(-1)}}} using the associative property



{{{A^2=XD(I)DX^(-1)}}} since {{{X*X^(-1)=I}}}



{{{A^2=XD*DX^(-1)}}}



{{{A^2=XD^2X^(-1)}}}



{{{A^2=XDX^(-1)}}} because from part a) we proved that {{{D^2=D}}}



{{{A^2=A}}} Using the definition that {{{A=XDX^(-1)}}}



So since we've shown that {{{A^2=A}}}, this means that matrix A is idempotent.



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