Compare eigenvalues of comparable matrices

Question

Consider two symmetric matrices $A,B$ on $\mathbb{R}^n$ and assume that $A\geq B$ as bilinear forms, i.e. $A(x,x)\geq B(x,x)$ for any vector $x$. Let $(\lambda_i)$ and $(\mu_i)$ be the ordered (possibly repeated with multiplicity) eigenvalues of $A$ and $B$, respectively. Is it true that $\lambda_i\geq\mu_i$ for every $i=1,...,n$? The result is obviously true when I can diagonalize the matrices simultaneously, but I am not sure what I can say in the general case.

Answer 1

It is true in general by the Courant-Fischer min-max theorem.