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In a paper from 1936, Harold Hotelling (access on JSTOR) defined the concepts of canonical correlations and canonical variates for two sets of variates. In pages 327 and 328, he precisely derives these two concepts with an algebraic perspective. In page 328, he deduces that if one set has more variables than the other $t>s$, then at least $t-s$ canonical variates are 0. This becomes clear in the subsequent pages, but I fail to grasp the following argument:

Then of the $s+t$ roots at least $t-s$ vanish; for the coefficients of $λ^{t-s-1}$ and lower powers of $λ$ are sums of principal minors of $2s+1$ or more rows, in which $λ$ is replaced by zero, and every such minor vanishes, as can be seen by a Laplace expansion.

Specifically, I do not see the vanishing by Laplace expansion. From my understanding, this is a Laplace expansion by complementary minors. Should we be specific in picking the $t-s-1$ (or less) rows?

Glen_b
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M. Toya
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  • Although it is a question about a statistical analysis procedure, it seems to be more mathematical than statistical. It is not about the algebra of CCA algoritm (such as outlined [here](http://stats.stackexchange.com/q/77287/3277) and in other threads here), but rather is about linear algebra. Maybe (perhaps) it is better to move the question to a math site? – ttnphns Mar 03 '17 at 12:27
  • @ttnphns The context probably is important. At least to arouse interest. – M. Toya Mar 03 '17 at 14:47

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