### Abstract

*z*, and arise e.g. as cross spectral density matrices in broadband array problems. Specifically, conditions for the existence and uniqueness of eigenvalues and eigenvectors of a parahermitian matrix EVD are given, such that these can be represented by a power or Laurent series that is absolutely convergent, at least on the unit circle, permitting a direct realisation in the time domain. Based on an analysis on the unit circle, we prove that eigenvalues exist as unique and convergent but likely infinite-length Laurent series. The eigenvectors can have an arbitrary phase response, and are shown to exist as convergent Laurent series if eigenvalues are selected as analytic functions on the unit circle, and if the phase response is selected such that the eigenvectors are Hölder continuous with α>½ on the unit circle. In the case of a discontinuous phase response or if spectral majorisation is enforced for intersecting eigenvalues, an absolutely convergent Laurent series solution for the eigenvectors of a parahermitian EVD does not exist. We provide some examples, comment on the approximation of a parahermitian matrix EVD by Laurent polynomial factors, and compare our findings to the solutions provided by polynomial matrix EVD

algorithms.

Language | English |
---|---|

Pages | 2659-2672 |

Number of pages | 14 |

Journal | IEEE Transactions on Signal Processing |

Volume | 66 |

Issue number | 10 |

Early online date | 6 Mar 2018 |

DOIs | |

State | Published - 15 May 2018 |

### Fingerprint

### Keywords

- hermitian matrix
- parahermitian matrix
- eigenvalue decomposition algorithm

### Cite this

}

*IEEE Transactions on Signal Processing*, vol. 66, no. 10, pp. 2659-2672. DOI: 10.1109/TSP.2018.2812747

**On the existence and uniqueness of the eigenvalue decomposition of a parahermitian matrix.** / Weiss, Stephan; Pestana, Jennifer; Proudler, Ian K.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the existence and uniqueness of the eigenvalue decomposition of a parahermitian matrix

AU - Weiss,Stephan

AU - Pestana,Jennifer

AU - Proudler,Ian K.

PY - 2018/5/15

Y1 - 2018/5/15

N2 - This paper addresses the extension of the factorisation of a Hermitian matrix by an eigenvalue decomposition (EVD) to the case of a parahermitian matrix that is analytic at least on an annulus containing the unit circle. Such parahermitian matrices contain polynomials or rational functions in the complex variable z, and arise e.g. as cross spectral density matrices in broadband array problems. Specifically, conditions for the existence and uniqueness of eigenvalues and eigenvectors of a parahermitian matrix EVD are given, such that these can be represented by a power or Laurent series that is absolutely convergent, at least on the unit circle, permitting a direct realisation in the time domain. Based on an analysis on the unit circle, we prove that eigenvalues exist as unique and convergent but likely infinite-length Laurent series. The eigenvectors can have an arbitrary phase response, and are shown to exist as convergent Laurent series if eigenvalues are selected as analytic functions on the unit circle, and if the phase response is selected such that the eigenvectors are Hölder continuous with α>½ on the unit circle. In the case of a discontinuous phase response or if spectral majorisation is enforced for intersecting eigenvalues, an absolutely convergent Laurent series solution for the eigenvectors of a parahermitian EVD does not exist. We provide some examples, comment on the approximation of a parahermitian matrix EVD by Laurent polynomial factors, and compare our findings to the solutions provided by polynomial matrix EVD algorithms.

AB - This paper addresses the extension of the factorisation of a Hermitian matrix by an eigenvalue decomposition (EVD) to the case of a parahermitian matrix that is analytic at least on an annulus containing the unit circle. Such parahermitian matrices contain polynomials or rational functions in the complex variable z, and arise e.g. as cross spectral density matrices in broadband array problems. Specifically, conditions for the existence and uniqueness of eigenvalues and eigenvectors of a parahermitian matrix EVD are given, such that these can be represented by a power or Laurent series that is absolutely convergent, at least on the unit circle, permitting a direct realisation in the time domain. Based on an analysis on the unit circle, we prove that eigenvalues exist as unique and convergent but likely infinite-length Laurent series. The eigenvectors can have an arbitrary phase response, and are shown to exist as convergent Laurent series if eigenvalues are selected as analytic functions on the unit circle, and if the phase response is selected such that the eigenvectors are Hölder continuous with α>½ on the unit circle. In the case of a discontinuous phase response or if spectral majorisation is enforced for intersecting eigenvalues, an absolutely convergent Laurent series solution for the eigenvectors of a parahermitian EVD does not exist. We provide some examples, comment on the approximation of a parahermitian matrix EVD by Laurent polynomial factors, and compare our findings to the solutions provided by polynomial matrix EVD algorithms.

KW - hermitian matrix

KW - parahermitian matrix

KW - eigenvalue decomposition algorithm

UR - http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=78

UR - http://ieeexplore.ieee.org/document/8307197/

U2 - 10.1109/TSP.2018.2812747

DO - 10.1109/TSP.2018.2812747

M3 - Article

VL - 66

SP - 2659

EP - 2672

JO - IEEE Transactions on Signal Processing

T2 - IEEE Transactions on Signal Processing

JF - IEEE Transactions on Signal Processing

SN - 1053-587X

IS - 10

ER -