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Operator Reverse Monotonicity of the Inverse

Alexis Akira Toda
The American Mathematical Monthly
Vol. 118, No. 1 (January 2011), pp. 82-83
DOI: 10.4169/amer.math.monthly.118.01.082
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Page Count: 2
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Abstract In statistics and econometrics, the equivalence between matrix inequalities A ≥ B ⇔ B−1 ≥ A−1 is used to obtain a lower bound on the variance matrix, where A, B are symmetric and positive definite. The same property holds for linear operators on Hilbert spaces that are bijective, self-adjoint, and positive definite. I give a short and elementary proof of this fact.

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