On a Characterization of the Best l2 Scaling of a Matrix

This paper is concerned with best two-sided scaling of a general square matrix, and in particular with a certain characterization of that best scaling: namely that the first and last singular vectors (on left and right) of the scaled matrix have components of equal modulus, necessity, sufficiency, and its relation with other characterizations are discussed. Then the problem of best scaling for rectangular matrices is introduced and a conjecture made regarding a possible best scaling. The conjecture is verified for some special cases.