# Trace minimization for generalized eigenvalue problem

In , 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 ), 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 :)