In what sense is the weighted Frobenius norm adimensional/scale-invariant for any symmetric positive definite weight matrix $W$? If we plug in a positive diagonal matrix into $W$ wee see that $||A||_W$ is not always $||TA||_W$ for a one-to-one transformation $T$. See page 196 of Nocedal's book on numerical optimization and page 60 of Fletcher's Practical methods of numerical optimization for reference.
Weighted Frobenius norm in BFGS
optimization;quasi newton