Lecture 19

Abstract

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If $f(x) = \tfrac{1} {2}(Ax,x) - \operatorname{Re} (b,x), A = A^* \in \mathbb{C}^{n \times n}$, then the boundedness of f from below is equivalent to the nonnegative definiteness of A (prove this). Let us assume that A > 0. In this case, a linear system Ax = b has a unique solution z, and, for any x,

$$ f(x) - f(z) = \frac{1} {2}(A(x - z),x - z) \equiv E(x). \Rightarrow $$

z is the single minimum point for f (x). ⇒ A minimization method for f can equally serve as a method of solving a linear system with the Hermitian positively definite coefficient matrix.