In [[1](https://www.jstor.org/stable/pdf/2157207.pdf?refreqid=excelsior%3Ac8cb5873955a58f5a2eba65be8b3f13c)], it is shown in theorem 1.2 that for symmetric $n \times n$ matrices $A$, $B$, we have
$$
\min\_{Y \in Y^\*} \text{tr}(Y^TAY) = 
\text{tr}(X^TAX) = 
\sum\_{i=1}^p \lambda\_i,
$$
with
$$
\text{
$X^TBX = I^p$
and 
$X^TAX = \mathrm{diag}(\lambda\_1,\dots,\lambda\_p)$,
}
$$  

$Y^\*$ being the set of all $n\times p$ matrices for which $Y^TBY = I^p$, and the columns of $X$ correspond to the eigenvectors belonging to the first $p$ eigenvalues $\lambda\_1 \leq \dots \leq \lambda\_p$.


It is then stated that this is a direct consequence of theorem 1.1 above and the Courant-Fischer theorem. While I understand the proof for 1.1 (as given in [[2]](https://books.google.nl/books?id=eXQXwCDD9agC&lpg=PR2&ots=AiStMDbESI&dq=J.%20N.%20FRANKLIN%2C%20Matrix%20Theory%2C%20Prentice%20Hall%2C%20Englewood%20Cliffs&lr&hl=nl&pg=PA106#v=onepage&q=J.%20N.%20FRANKLIN,%20Matrix%20Theory,%20Prentice%20Hall,%20Englewood%20Cliffs&f=false)), I don't get how this is used together with C-F to yield the proof for 1.2. A lot of other papers refer to this proof, so that doesn't help figuring it out. Moreover, I read that the min-max theorem and the Cauchy interlacing theorem both get referred to as Courant-Fischer theorem sometimes, so I'm note sure which C-F the writers are referring to.


I'd be very thankful for any help or suggestions :)



Trace minimization for generalized eigenvalue problem
After some work, I figured out the proof, using -- indeed -- the Courant-Fischer theorem and parts of Cauchy's Interlacing and Poincaré's Separation theorems. I've carved out the part of my thesis proving this, which can be found [here](https://github.com/dnndbrkt/stratifyingautism/blob/6e4660b66c351cb83177736e963752e9d9f2c13f/th%203.8.pdf).


