Kantorovich inequality

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Let LaTeX: Q be a symmetric, positive definite matrix, with eigenvalues in LaTeX: [a,b], and let LaTeX: x \in \mathbb{R}^n. Then,

LaTeX: 
\frac{\| x \|^2}{(x^T Q x)(x^T Q^{-1}x)} \le \frac{4 \, a \, b}{(a+b)^2}.

Its significance in mathematical programming is in convergence analysis: it bounds the convergence rate of Cauchy's steepest ascent.


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