Rank and Inertia Theorems for Matrices: The Semi-definite Case |
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a₁ A₂ assume H b₁ b₂ bounds on rank c₁ chains chapter containing x terms Corollary corresponding d₁ d₂ decomposition define H direct sum divisors of degrees divisors of imaginary e₁ elementary divisor structure exists a non-singular exists an H Gives bounds H conformably H Hij H is non-singular H₁ H₁₁ H₁₂ hence Hermitian matrix iiii imaginary roots induction inequalities inertia triple integer Jordan canonical form k₁ Lemma Let H lower bound Lyapunov Main Inertia Theorem Math matrix H n x n matrix non-imaginary roots non-zero elements P₁ partitioned conformably proof of Theorem rank H rank of H rank R(AH Reduction rows and columns standard notation submatrix superdiagonal Suppose Theorem 4.5 thesis U₁ v₁ X X X X X X Y₁ zero ΣΙΣ ра