Summary
A version of Cauchy's inequality is obtained which relates two matrices by an inequality in the sense of the Loewner ordering. In that ordering a symmetric idempotent matrix is dominated by the identity matrix and this fact yields a simple proof.
A consequence of this matrix Cauchy inequality leads to a matrix version of the Kantorovich inequality, again in the sense of Loewner.
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Marshall, A.W., Olkin, I. Matrix versions of the Cauchy and Kantorovich inequalities. Aeq. Math. 40, 89–93 (1990). https://doi.org/10.1007/BF02112284
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DOI: https://doi.org/10.1007/BF02112284