Condition number

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This is LaTeX: \|A\| \|A^{-1}\| when LaTeX: A is nonsingular and LaTeX: \|\cdot\| is some matrix norm. This arises in convergence analysis, where LaTeX: A is the hessian. Whenever LaTeX: A is symmetric and positive definite (as the hessian of a strictly convex function), and the matrix norm is that induced by the Euclidean norm (i.e., LaTeX: \|A\| := \max \{\|Ay\|: \|y\|=1\}), the condition number is the ratio of the extreme values of the eigenvalues. It is often used to measure the convergence rate of an ascent algorithm, due to Kantorovich's inequality. (Unfortunately, it is erroneously used to measure the goodness of an algorithm – see Myths and Counterexamples in Mathematical Programming.)

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