How is the orthogonal projection on to the span of the columns of a matrix determined by a chosen inner product?

Question

I know that of course a orthogonal projection must be orthogonal for a chosen inner product.

But how can I find a new orthogonal projection based on $P=A(A^TA)^{-1}A^T$, if I have dot product defined as $(x,y)=x^TMy$?

Answer 1

The adjoint $A^*$ of $A$ relative to an inner product is defined by $$ (Ax,y) = (x,A^*y) \quad \forall x,y $$ For the usual inner product, we have $A^* = A^T$. For this new inner product, we need to have $$ (Ax)^T M y = x^T M (A^*y) \implies\\ x^T A^TM y = x^T M A^* y \implies\\ A^TM = MA^* \implies\\ A^* = M^{-1}A^T M $$ We then have $$ P = A(A^*A)^{-1}A^* = A(M^{-1}A^T M A)^{-1} (M^{-1}A^T M) =\\ A(A^T M A)^{-1} A^T M $$