Computing the closest real normal matrix and normal completion

In this article, we consider the problems (unsolved in the literature) of computing the nearest normal matrix X to a given non-normal matrix A , under certain constraints, that are (i) if A is real, we impose that also X is real; (ii) if A has known entries on a given sparsity pattern Ω and unknown/uncertain entries otherwise, we impose to X the constraint x i j = a i j for all entries ( i , j ) in the pattern Ω . As far as we know, there do not exist in the literature specific algorithms aiming to solve these problems. For the case in which all entries of A can be modified, there exists an algorithm by Ruhe, which is able to compute the closest normal matrix. However, if A is real, the closest computed matrix by Ruhe’s algorithm might be complex, which motivates the development of a different algorithm preserving reality. Normality is characterized in a very large number of ways; in this article, we consider the property that the square of the Frobenius norm of a normal matrix is equal to the sum of the squares of the moduli of its eigenvalues. This characterization allows us to formulate as equivalent problem the minimization of a functional of an unknown matrix, which should be normal, fulfill the required constraints, and have minimal distance from the given matrix A .