The Normal Defect of Some Classes of Matrices

TL;DR

This paper investigates the normal defect of specific classes of matrices, providing explicit calculations for 4x4 matrices with superdiagonal entries and extending results to certain n x n matrices, including block diagonal cases.

Contribution

It computes the normal defect for a special class of 4x4 matrices and establishes conditions for the normal defect of block diagonal matrices, including examples where the defect is not additive.

Findings

Normal defect of 4x4 superdiagonal matrices explicitly computed

Constructed minimal normal completion matrices for these classes

Identified conditions where the normal defect of block diagonal matrices equals the sum of defects

Abstract

An n \times n matrix A has a normal defect of k if there exists an (n+k) \times (n+k) normal matrix A_{ext} with A as a leading principal submatrix and k minimal. In this paper we compute the normal defect of a special class of 4 \times 4 matrices, namely matrices whose only nonzero entries lie on the superdiagonal, and we provide details for constructing minimal normal completion matrices A_{ext}. We also prove a result for a related class of n \times n matrices. Finally, we present an example of a 6 \times 6 block diagonal matrix having the property that its normal defect is strictly less than the sum of the normal defects of each of its blocks, and we provide sufficient conditions for when the normal defect of a block diagonal matrix is equal to the sum of the normal defects of each of its blocks.