This is my first post and I was wondering whether any of you can help me out?
Consider the n x k matrix X, and the projection matrix Px = X ((XTX)^1) XT
where XT= is the X transpose and Px is a term on its own NOT P*x
a) What are the dimensions of Px, (number of rows and columns)?
b) Prove PxPx = Px.
c) Does Py = Px if Y = XA, where A is an k x k nonsingular matrix?
I was able to solve a) and b)...I think...dimensions of Px=nxn and Px=I (I still have some trouble with this)
However I am having trouble solving c)
Can anyone help?
I'm having a bit of trouble reading what you typed, is this accurate:
Why is there a to the 1 power in there? What are the lower-case x and y? How are they connected to big X and Y?
Hi and thanks,
the bracket is to the power of -1. and according to the question, Px is defined as the Projection matrix. Px is a term in itself, it is not P * x . As for relation between small x and y, I can't seem to figure out anything else past what the question mentions?
do you think it's correct if I say
if (XT,X)^-1 exists,
then the product of X transpose times X has an existing inverse,
hence det(XT*X) does not equal 0,
and that also means det(XT) and det(X) do not equal zero
so therefore XT and X are square matrices, are invertible and their dimensions nxk, is actually, nxk where n=k?
is this correct?