Matrix Norm With Respect to Another Matrix?

Question

I'm reading through a book on numerical analysis right now and in its treatment of linear iterative method the author takes the norm of a  matrix with respect to another matrix but never seems to say what this actually means.
For a little more context we are dealing with iterative methods of the form $x^{(k + 1)} = Bx^{(k)} + f$ to solve $Ax = b$, then splitting $A = P - N$ where $P$ is suitably nice and $B = P^{-1}N$. Then the author makes guarantees on the spectral radius of $B$ given certain criteria on $A$ and $P$ and states during this:
$$rho(B) = Vert B Vert_A = Vert B Vert_P < 1$$
Perhaps it is the induced norm of $Vert cdot Vert_A$, ie.
$$Vert B Vert_A = sup_{Vert x Vert_A = 1} frac{(Bx)^TA(Bx)}{x^TAx}?$$
What does $Vert cdot Vert_A$ with respect to another matrix mean?

Lutz Lehmann · Answer

This is called an "elliptical" norm, or the Euclidean norm relative to the scalar product defined by $A$. Which obviously demands that $A$ is s.p.d.
The unit sphere of the $A$ norm is an ellipsoid in standard Euclidean coordinates, thus the name.
Note that your formula defines $|B|_A^2$.

Matrix Norm With Respect to Another Matrix?

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