Symmetric positive definite with respect to an inner product Let $A$ be a SPD(symmetric positive-definite) real $n\times n$ matrix. let $B=LL^T$ be also SPD. Let $(,)_B$ be an inner product given by $(x,y)_B=x \cdot By=y^T Bx$. Then $(B^{-1}Ax,y)_B=(x,B^{-1}Ay)_B$ for all $x,y$. Show that $B^{-1}A$ is SPD with respect to $(,)_B$
I don't understand what it means SPD with respect to an inner product. What does it mean?

It means that in the definition of positive definite matrix you replace a standard euclidean scalar product by an inner product generated by another matrix.
A matrix $C$ is positive definite with respect to inner product $(,)_B$ defined by a positive definite matrix $B$ iff
$$\forall v\ne 0\, (Cv,v)_B = (BCv,v)>0$$
or, in our case
$$\forall v\ne 0\, (B^{-1}Av,v)_B = (Av,v)>0,$$ which is given.
