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Yasuhiko Ikebe, Toshiyuki Inagaki, Sadaaki Miyamoto, The Monotonicity Theorem, Cauchy's Interlace Theorem, and the Courant- Fischer Theorem, The American Mathematical Monthly, Vol. 94, No. 4 (Apr., 1987), pp. 352-354
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Ikebe et al. (1987) studied this question.
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