3.4 Newton’s Method With Hessian Modification

Definition inertia

• $A \in \mathbb{R}^{n \times n}$,
  • symmetric matrix
• $\lambda_{i}(A)$,
  • $i$-th leargest eigenvalues

Let

\[\begin{eqnarray} n_{+}(A) & := & \mathrm{card}( \{ i = 1, \ldots, n \mid \lambda_{i}(A) > 0 \} ) \nonumber \\ n_{-}(A) & := & \mathrm{card}( \{ i = 1, \ldots, n \mid \lambda_{i}(A) < 0 \} ) \nonumber \\ n_{0}(A) & := & \mathrm{card}( \{ i = 1, \ldots, n \mid \lambda_{i}(A) = 0 \} ) \nonumber \end{eqnarray}\]

Inertia of a matrix $A$ is a triplet \(\mathrm{inertia}(A) := (n_{+}, n_{-}, n_{0})\).

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