Recent content by megalomaniac

Wow I'm an imbecile. I skipped over that fact and just focused on trying to prove which singular matrices would fit the bill, but this was not required. Thank you so much for your guidance! I learned more than what the proof expected of me.

I was thinking, since I've proven the only non-singular idempotent matrix is the identity matrix, wouldn't it follow that the only other idempotent matrices must be singular?

Oh wow! I never looked at it that way! How does it follow that A has determinant 0? -Thanks!

But I'm trying to prove that if an idempotent matrix is not the identity matrix, then it must be singular. How do I make this leap? -Thank you!

But how does that prove that A must be singular?

Ha oops, How about: P-1APP-1AP = P-1AIAP = P-1AAP = P-1AP. So B2 = B ?

So B2=(P-1)2A2P2 = AI = A?

I haven't spent too much time dealing with similar matrices, but I'm assuming I am to use the fact that if B=P-1AP for some invertible matrix P, then A is similar to B. And A would be an idempotent diagonal matrix in this case?

So am I to show that a singular diagonal idempotent matrix is similar to a singular non-diagonal idempotent matrix? I'm sorry, I'm still unsure on how to go about this, my mind's been all over the place.

Ok, I have determined that detA equals the product of the eigenvalues. So in order for A to be singular, at least one eigenvalue must be zero. Where would I go from here to prove a singular idempotent matrix must have at least one zero eigenvalue? -Thanks!

So far it seems that the sum of the diagonal entries of an idempotent singular matrix must equal one, and the rows and/or columns must be linearly dependent.

So for all diagonal idempotent matrices, A2 = A iff ai2 = ai for all i=1,...,n. This can only occur for i values of 0 and 1. Therefore, a diagonal matrix is idempotent iff each diagonal entry is 0 or 1. So any idempotent diagonal matrix will be singular (save for I), but this still doesn't...

If A is idempotent and non-singular: If I premultiply both sides by A-1, then I get: A = InA = A-1AA = A-1A = In. Would this be valid? As for the diagonal idempotent matrices, they have full rank and therefore are non-singular and symmetric and non-diagonal idempotent matrices would...

Isn't the only diagonal idempotent matrix the identity matrix?

Thus far I have shown, Suppose A is an idempotent nxn matrix. In the case of a non-singular matrix, in order to be invertible, there exists an nxn matrix B such that AB = BA = In, where B = A-1 Since A = B and B = A-1, then A = A-1 AA = In Therefore In is idempotent. Would this...