For a given square matrix A the predicted values of matrix B are:

Question

predicted B=A(A'A)^(-1)A'B

why is the matrix C=A(A'A)^(-1)A' an idempotent and symmetric matrix? and is this matrix invertible?

Answer 1

Assuming (A'A) is invertible, then (A'A)^-1 exists.

A(A'A)^-1A'
By the property of inverse of product of matrices,
(A'A)^-1
=A^-1 A'^-1

Therefore
C=A(A'A)^-1A'
=A(A^-1 A'^-1)A'
=(A A^-1) (A'^-1A')
= (I) (I)
=I
after application of associativity and the properties of inverse of matrices.
Since I is idempotent and invertible, so is C.

Answer 2

The equation of the parabola which contains 2 points (1,1) and (-2,-2) and whose tangent at the point (1,1) has the slope k is y=1/3(A)x^2+1/3(B)x-2/3(C) Express A,B and C with k.Please solve this problem.please