Squared norms: difference or notational convenience
Given certain matrix [tex]A\in\mathbb{R}^{n\times m},[/tex]the rank d approximation L with the same number of rows/column as A, minimizing the Frobenius norm of the difference [tex]AL[/tex] is matrix obtained by singular value decomposition of A, with only d dominant singular values (the rest is simply set to zero).
However, I often encounter the minimization of the adapted norm, such as various kinds of normalization on the norm, ie. [tex]i) AK^2[/tex] [tex]ii) \left(\frac{AK}{A}\right)^{1/2}[/tex] and I'm not sure if the solution L from the above nonsquared Frobenius norm coincides with the normalized Frobenius norm solution from i) and ii). Isn't it the case that K should be L, but appropriately scaled for i) and/or ii)? 
Re: Squared norms: difference or notational convenience
Essentially you're given a function f(K) and asked to minimize it. You're then asked to minimize f(K)^2 and f(K)/constant. All of these functions have the same minimum because the operations you are applying to f are all monotone

Re: Squared norms: difference or notational convenience
Thanks; I had similar reasoning. However, I'm surprised that in the literature one might find some confusing monotone transformations.

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