Yes, this follows from the Maximin characterization of eigenvalues of symmetric matrices. If $A$ is an $n\times n$ symmetric matrix, form the Rayleigh ratio $R(x)=(x,Ax)/(x,x),$ where $(.,.)$ is the standard dot product. Then the $k$-th eigenvalue is $$\lambda_k=\max_a\min_{ax=0} R(x)$$ where the inner minimum is under the condition that we impose $k-1$ linear restrictions ($a$ is a $(k-1)\times n$ matrix) on $x$, and the outer maximum is over all possible restrictions.

If you add to $A$ a positive-definite matrix, the Rayleigh ratio evidently increases, from which the result follows.

Yes, this follows from the Maximin characterization of eigenvalues of symmetric matrices. If $A$ is an $n\times n$ symmetric matrix, form the Rayleigh ratio $R(x)=(x,Ax)/(x,x),$ where $(.,.)$ is the standard dot product. Then the $k$-th eigenvalue is $$\lambda_k=\max_a\min_{ax=0} R(x)$$ where the inner minimum is under the condition that we impose $k-1$ linear restrictions ($a$ is a $(k-1)\times n$ matrix) on $x$, and the outer maximum is over all possible restrictions.

If you add to $A$ a positive-definite matrix, the Rayleigh ratio evidently increases, from which the result follows.

Ref. R. Gantmacher and M. Krein, Oscillation matrices... AMS 2002, or any textbook on classical mechanics, for example V. Arnold, Mathematical methods of classical mechanics.