### Abstract

It is proven that a necessary and sufficient condition for an n-dimensional Hermitian matrix to be positive definite is that it be expressible in the form =OEO, where O is a complex orthogonal matrix and E is a diagonal matrix with positive elements. This accomplishes a parametrization since O has n2-n real parameters and E has n of them. The proof is constructive, giving O and E. It is further shown that the limit forms of this expression yield all the non-negative definite matrices. The parametrization for the polarization matrix of a spin-one particle is given explicitly.

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

Journal | Physical Review |

Volume | 136 |

Issue number | 2B |

DOIs | |

State | Published - 1964 |

### Fingerprint

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

**Simple theorem on Hermitian matrices and an application to the polarization of vector particles.** / Zwanziger, Daniel.

Research output: Contribution to journal › Article

*Physical Review*, vol. 136, no. 2B. https://doi.org/10.1103/PhysRev.136.B558

}

TY - JOUR

T1 - Simple theorem on Hermitian matrices and an application to the polarization of vector particles

AU - Zwanziger, Daniel

PY - 1964

Y1 - 1964

N2 - It is proven that a necessary and sufficient condition for an n-dimensional Hermitian matrix to be positive definite is that it be expressible in the form =OEO, where O is a complex orthogonal matrix and E is a diagonal matrix with positive elements. This accomplishes a parametrization since O has n2-n real parameters and E has n of them. The proof is constructive, giving O and E. It is further shown that the limit forms of this expression yield all the non-negative definite matrices. The parametrization for the polarization matrix of a spin-one particle is given explicitly.

AB - It is proven that a necessary and sufficient condition for an n-dimensional Hermitian matrix to be positive definite is that it be expressible in the form =OEO, where O is a complex orthogonal matrix and E is a diagonal matrix with positive elements. This accomplishes a parametrization since O has n2-n real parameters and E has n of them. The proof is constructive, giving O and E. It is further shown that the limit forms of this expression yield all the non-negative definite matrices. The parametrization for the polarization matrix of a spin-one particle is given explicitly.

UR - http://www.scopus.com/inward/record.url?scp=36149010367&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=36149010367&partnerID=8YFLogxK

U2 - 10.1103/PhysRev.136.B558

DO - 10.1103/PhysRev.136.B558

M3 - Article

AN - SCOPUS:36149010367

VL - 136

JO - Physical Review

JF - Physical Review

SN - 0031-899X

IS - 2B

ER -