Properties of the Approximations of a Matrix which Lower its Rank

Let C be a real m ? n matrix. This paper considers the generalproperties of the best approximations E to C in an arbitrarynorm by matrices of rank r. The case in which rank C = n, r= n – 1, and ||?|| is a separable norm is dealt with separately.For this particular case, the form of every best approximationE and an estimate of the error ||C – E|| are given. Thepaper also investigates when, for this particular case, thebest approximation E is unique.     

Properties of the {ell}1-solutions of the Linear Matrix Equation AX+YB=C

The properties of the ₁ of the linear matrix equation AX+YB=Care investigated, where A, B, and C are given real matricesof dimensions m x r, s x n, and m x n, respectively, with m> r and n > s. An algorithm which is an ₁version of thegeneral alternating method is developed. This algorithm utilizesa special form of the equation AX+YB=C and adjusts X and Y alternately.It gives an ₁-solution of AX+YB=C under additional assumptions.Some numerical examples are given.