## Inequalities for Sign Frequencies of Latent RootsContents: Notation Inequalities for Gramian matrices Inequalities for Hermitian matrices Inequaltiies for the sum of Hermitian matrices Inequalities for products The general case. |

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### Contents

k Inequalities for Hermitian Matrices | 6 |

Inequalities for the Sum | 10 |

Inequalities for Products | 13 |

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absolute value adding a Gramian Applied Social Research Applying this right coefficients of imaginary Comparing 31 Corollary diagonal elements diagonal matrix defined ensue the identities equality holding establish Theorem exist corresponding roots follows from 26 FREQUENCIES OF LATENT Furthermore G is Gramian Gramian matrix H is Hermitian H2 shows Hebrew University Hermitian and Gramian Hermitian H Hermitian matrices INEQUALITIES FOR SIGN inequality of 30 inspection of 32 Institute of Applied Israel Institute law of inertia left inequality Lemma Louis Guttman main diagonal n^XHX negative latent roots non-vanishing nonsingular matrices normal matrix nQ(A number of negative number of positive nx(G partitioned matrix polar form positive latent roots proof r(DA real and non-negative reduce the number Research Technical Note right inequality roots of GH roots of H SIGN FREQUENCIES SOCIAL RESEARCH JERUSALEM Social Research Technical square matrix Sylvester's law uniquely defined unitary matrix whence the left