Minimal Normal and Commuting Completions

We study the minimal normal completion problem: given A ∈ Cn×n, how do we find an (n+q)×(n+q) normal matrix Aext := ( A A12 A21 A22 ) of smallest possible size? We will show that this smallest number q of rows and columns we need to add, called the normal defect of A, satisfies nd(A) ≥ max{i−(AA∗ −A∗A), i+(AA∗ −A∗A)}, where i±(M) denotes the number of positive and negative eigenvalues of the Hermitian matrix M counting multiplicities. Subsequently, we will show that for some matrices a minimal normal completion can be chosen to be a multiple of a unitary, addressing a conjecture from [H. J. Woerdeman, Linear and Multilinear Algebra 36 (1993), 59–68]. In addition, we study the related question where A ∈ Cn×n and B ∈ Cn×n are given, and where we look for Aext := ( A A12 A21 A22 ) and Bext := ( B B12 B21 B22 ) such that they commute and are of smallest possible size.