Eigenvalues of $MA$ versus eigenvalues of $A$ for orthogonal projection $M$

Solution 1:

As $M$ is symmetric idempotent, with respect to its orthonormal eigenbasis, you may assume that $M=I_{n-k}\oplus0$. Then the eigenvalues of $MA$ are just the eigenvalues of the leading principal $(n-k)\times(n-k)$ submatrix of $A$. So, essentially, the inequality in question relates the eigenvalues of a positive definite matrix $A$ to the eigenvalues of its principal submatrix. This actually is a well-known result that is not only true for positive definite matrices, but for all Hermitian matrices. See, e.g. theorem 4.3.15 (p.189) of Horn and Johnson, Matrix Analysis, 1/e, Cambridge University Press, 1985.