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assume H best possible bounds bounds on rank chains chapter completes the proof Corollary corresponding define H determine direct sum direct summand divisors of degrees divisors of imaginary DOCTOR OF PHILOSOPHY elementary divisor structure exists a non-singular exists an H Gives bounds H is non-singular hence Hermitian matrix i=l i=l i=l j i=M+l j induction inequalities inertia triple integer ir(A JJ JJ Jordan canonical form Let H lower bound Lyapunov Main Inertia Theorem Math matrix H matrix theory n x n matrix non-imaginary roots non-zero elements non-zero minor number of elementary partition H conformably pq pq proof of Theorem qq qq R(A H rank H rank of H rank R(AH Reduction rows and columns satisfying 2.15 standard notation submatrix superdiagonal term Theorem 6.1 thesis tt,v tt(A tt(H University of Wisconsin zero