Journal article
MATRICES WITH NORMAL DEFECT ONE
OPERATORS AND MATRICES, Vol.3(3), pp.401-438
01 Sep 2009
Featured in Collection : UN Sustainable Development Goals @ Drexel
Abstract
A n x n matrix A has normal defect one if it is not normal, however can be embedded as a north-western block into a normal matrix of size (n + 1) x (n + 1). The latter is called a minimal normal completion of A. A construction of all matrices with normal defect one is given. Also, a simple procedure is presented which allows one to check whether a given matrix has normal defect one, and if this is the case - to construct all its minimal normal completions, A characterization of the generic case for each n under the assumption rank (A*A - AA*) = 2 (which is necessary for A to have normal defect one) is obtained. Both the complex and the real cases are considered. It is pointed out how these results can be used to solve the minimal commuting completion problem in the classes of pairs of n x n Hermitian (resp., symmetric, or symmetric/antisymmetric) matrices when the completed matrices are sought of size (n + 1) x (n + 1). An application to the 2 x n separability problem in quantum computing is described.
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Details
- Title
- MATRICES WITH NORMAL DEFECT ONE
- Creators
- Dmitry S. Kaliuzhnyi-Verbovetskyi - Drexel Univ, Dept Math, Philadelphia, PA 19104 USAIlya M. Spitkovsky - Coll William & Mary, Dept Math, Williamsburg, VA 23187 USAHugo J. Woerdeman - Drexel University
- Publication Details
- OPERATORS AND MATRICES, Vol.3(3), pp.401-438
- Publisher
- Element
- Number of pages
- 38
- Grant note
- DMS-0456625; DMS-0500678 / NSF; National Science Foundation (NSF)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Identifiers
- 991019170554104721
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