How can one interpret a matrix norm like this $\| A \|_W := \|W^\frac{1}{2} A W^\frac{1}{2} \|_F$ for positive definite W?

Question

In our optimization class we derived the Davidon/Fletcher/Powell Formula using a 'weighted' Frobenius-Norm:

$\| A \|_W := \|W^\frac{1}{2} A W^\frac{1}{2} \|_F$

for a symmetric and positive definite matrix W. I have a really hard time understanding what this weighing does.

Is there a geometric interpretation here that eludes me?

Answer 1

Not a full answer, but perhaps more than a comment. The Frobenius norm is preserved under an orthogonal/unitary change of basis. Since $W$ is symmetric, it can be unitarily diagonalized, and this can be taken to be an orthogonal diagonalization if $W$ is real. Thus "it suffices" to make sense of what the right-hand-side looks like in a basis where $W$ (and hence $W^{\frac{1}{2}}$) is diagonal. I also recommend you consider the difference between multiplying by a diagonal matrix on the left vs. on the right -- how do they scale the matrix $A$?