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

Daniel Zwanziger

    Research output: Contribution to journalArticle

    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 languageEnglish (US)
    JournalPhysical Review
    Volume136
    Issue number2B
    DOIs
    StatePublished - 1964

    Fingerprint

    theorems
    polarization
    matrices

    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.

    In: Physical Review, Vol. 136, No. 2B, 1964.

    Research output: Contribution to journalArticle

    @article{48580910819c4e4eb62aafe5542a18c2,
    title = "Simple theorem on Hermitian matrices and an application to the polarization of vector particles",
    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.",
    author = "Daniel Zwanziger",
    year = "1964",
    doi = "10.1103/PhysRev.136.B558",
    language = "English (US)",
    volume = "136",
    journal = "Physical Review",
    issn = "0031-899X",
    publisher = "American Institute of Physics Publising LLC",
    number = "2B",

    }

    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 -