Rank and Inertia Theorems for Matrices: The Semi-definite Case |
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a₁ A₂ assume H b₁ b₂ bounds on rank c₁ chapter Corollary corresponding d₁ d₂ decomposition define H direct sum divisors of degrees divisors of imaginary elementary divisor structure exists a non-singular exists an H Gives bounds H Hij H is non-singular h pq H under R(AH H₁ H₁₁ H₁j hence Hermitian matrix iiii imaginary roots induction inequalities inertia triple integer Jordan canonical form K₁ Lemma Let H linear lower bound Lyapunov Main Inertia Theorem Math matrix H n x n matrix non-imaginary roots non-zero elements non-zero minor number of elementary P₁ partition H conformably proof of Theorem r₁ rank H rank of H rank R(AH Reduction rows and columns s₁ standard notation submatrix superdiagonal Suppose Theorem 4.5 thesis U₁ v₁ y₁ zero ΣΙΣ