On Fillmore's theorem extended by Borobia
Fillmore Theorem says that if A is an nxn complex non-scalar matrix and {\gamma}_1,...,{\gamma}_{n} are complex numbers with {\gamma}_1+...+{\gamma}_{n}=trA, then there exists a matrix B similar to A with diagonal entries {\gamma}_1,...,{\gamma}_{n}. Borobia simplifies this result and extends it to...
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| Main Authors | , |
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| Format | Journal Article |
| Language | English |
| Published |
16.04.2018
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| Subjects | |
| Online Access | Get full text |
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| Summary: | Fillmore Theorem says that if A is an nxn complex non-scalar matrix and
{\gamma}_1,...,{\gamma}_{n} are complex numbers with
{\gamma}_1+...+{\gamma}_{n}=trA, then there exists a matrix B similar to A with
diagonal entries {\gamma}_1,...,{\gamma}_{n}. Borobia simplifies this result
and extends it to matrices with integer entries. Fillmore and Borobia do not
consider the nonnegativity hypothesis. Here, we introduce a different and very
simple way to compute the matrix B similar to A with diagonal
{\gamma}_1,...,{\gamma}_{n}. Moreover, we consider the nonnegativity hypothesis
and we show that for a list {\Lambda}={{\lambda}_1,...,{\lambda}_{n}} of
complex numbers of Suleimanova or \v{S}migoc type, and a given list
{\Gamma}={{\gamma}_1,...,{\gamma}_{n}} of nonnegative real numbers, the
remarkably simple condition
{\gamma}_1+...+{\gamma}_{n}={\lambda}_1+...+{\lambda}_{n} is necessary and
sufficient for the existence of a nonnegative matrix with spectrum {\Lambda}
and diagonal entries {\Gamma}. This surprising simple result improves a
condition recently given by Ellard and \v{S}migoc in arXiv:.1702.02650v1. |
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| DOI: | 10.48550/arxiv.1804.05738 |