I was going through the concepts of orthogonal and oblique projections in linear algebra. It describes the projection of a vector $b$ onto a subspace. If $A$ is a given matrix, let its range define the subspace. Then for orthogonal projection with arbitrary basis vectors in $A$, the projector matrix is defined as $P=A(A^{*}A)^{-1}A^{*}$.
What I am really confused about is the range of Projector matrix P ($range$ P) and the range of the actual subspace ( defined by the column vectors in A, $range$ A). Are they identical? If not then what is the range of P?
Can you provide a geometric interpretation to it? Also I am starting my numerical linear algebra course, so can you guys provide some good resources for Projectors, Norms and Singular Value Decomposition?
Please clear my doubts, how elementary might it be to you.
Thank You.
Yes, $range(P) = range(A)$. From the definition of $P$, it's clear that $range(P) \subseteq range(A)$. For the other direction, if $y \in range(A)$, suppose $y = Ax$. Then $Py = A (A^* A)^{-1} A^* A x = A x = y$ so $y \in range(P)$.