The inner product of two column vectors v and w is defined as:
$$v \cdot w = w \cdot v = <v,w>=v^T w = <w,v>= w^T v =|proj_w(v)| = |proj_v(w)| = \text{a scalar value}$$
Now if I were to generalize this concept of Inner product between a Matrix A and column vector x, What do we call this type of inner product? and how do we define the order of the matrix in the inner product operator $<A,x>$ verses $<A,x>$...
do we write $A^T x$ as $<A,x>$
... or do we write $A^T x$ as $<x, A>$?
Does Projection of A onto x equal:
$$|proj_x(A)| = <A,x> / <A,A>$$
or does it equal:
$$|proj_x(A)| = <x,A> / <A,A>$$
Given Pseduo inverse:
$$x=Ay$$ $$A^Tx=A^TAy$$ $$y=\frac{A^Tx}{A^TA}$$
$$A^{\dagger} = \frac{A^T}{A^TA}$$
$$y=A^{\dagger}x$$
thus, pseudo inverse corresponds to an orthogonal projection of vector x onto the manifold of A...
can we say:
$$A^{\dagger}x = proj_x(A)$$?
or would that be:
$$A^{\dagger}x = proj_A(x)$$?