Algebraic proofs are given to some inequalities involving canonical correlation coefficients which, however, seem quite natural from statistical intuition. Though all of these inequalities can be easily verified from the well-known Courant-Fischer min-max theorem (c. f. Bellman [2], pp. 115-117), our method in this note seems to have some interests notwithstanding.
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Akihiro Nishi (1979) studied this question.
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