正规亏1矩阵
Matrices with normal defect one
摘要：
如果$n\times n$矩阵$A$不是正态的，则它具有正态缺陷1，然而可以作为西北块嵌入到大小为$(n+1)\times (n+1)$的正规矩阵中。后者称为$A$的最小正规完备化。给出了所有正规亏1矩阵的一种构造方法。此外，一个简单的程序，它允许一个检查是否有一个给定的矩阵正常亏损，如果是这种情况下-构造所有的最小正常完成。在假设$\rank(A^*A-AA^*)=2$（这是$A$具有正常缺陷1所必需的）下，得到了每个$n$的一般情况的表征。复杂的和真实的情况下被认为是。指出了如何利用这些结果来解决$n\times n$ Hermitian（分别为，对称或对称/反对称）矩阵。描述了量子计算中$2\times n$可分性问题的一个应用。
英文摘要：
A $n\times n$ 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 $(n+1)\times (n+1)$. 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 $n\times n$ Hermitian (resp., symmetric, or symmetric/antisymmetric) matrices when the completed matrices are sought of size $(n+1)\times(n+1)$. An application to the $2\times n$ separability problem in quantum computing is described.
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