### Abstract

A n×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)×(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×n Hermitian (resp., symmetric, or symmetric/antisymmetric) matrices when the completed matrices are sought of size (n+1)×(n+1). An application to the 2×n separability problem in quantum computing is described.

Original language | English (US) |
---|---|

Pages (from-to) | 401-438 |

Number of pages | 38 |

Journal | Operators and Matrices |

Volume | 3 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2009 |

### Fingerprint

### Keywords

- Commuting completions
- Complex symmetric operators and matrices
- Minimal normal completions
- Normal defect
- Separability problem
- Unitary and symmetric extensions

### ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory

### Cite this

*Operators and Matrices*,

*3*(3), 401-438. https://doi.org/10.7153/oam-03-24

**Matrices with normal defect one.** / Kaliuzhnyi-Verbovetskyi, Dmitry S.; Spitkovsky, Ilya; Woerdeman, Hugo J.

Research output: Contribution to journal › Article

*Operators and Matrices*, vol. 3, no. 3, pp. 401-438. https://doi.org/10.7153/oam-03-24

}

TY - JOUR

T1 - Matrices with normal defect one

AU - Kaliuzhnyi-Verbovetskyi, Dmitry S.

AU - Spitkovsky, Ilya

AU - Woerdeman, Hugo J.

PY - 2009/1/1

Y1 - 2009/1/1

N2 - A n×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)×(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×n Hermitian (resp., symmetric, or symmetric/antisymmetric) matrices when the completed matrices are sought of size (n+1)×(n+1). An application to the 2×n separability problem in quantum computing is described.

AB - A n×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)×(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×n Hermitian (resp., symmetric, or symmetric/antisymmetric) matrices when the completed matrices are sought of size (n+1)×(n+1). An application to the 2×n separability problem in quantum computing is described.

KW - Commuting completions

KW - Complex symmetric operators and matrices

KW - Minimal normal completions

KW - Normal defect

KW - Separability problem

KW - Unitary and symmetric extensions

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U2 - 10.7153/oam-03-24

DO - 10.7153/oam-03-24

M3 - Article

VL - 3

SP - 401

EP - 438

JO - Operators and Matrices

JF - Operators and Matrices

SN - 1846-3886

IS - 3

ER -