sum of positive definite matrix

Question

sum of positive definite matrix $A+B $is positive definite. I want to look at the spectrum of $C=A+B$

can we say the ith largest eigenvalue of $C$ is no less than the ith largest eigenvalue of $A$ i.e. $B$ as positive definite matrix, has contribution to the growth of spectrum?

remark:

sorry my question may be too stupid, actually it is originally from

sum of matrix and its spectrum

and

Sum of Gaussian matched by Brownian Motion?

very appreciated for the help!

if "sum of matrix and its spectrum" is solved, "Sum of Gaussian matched by Brownian Motion?" would be solved.

Answer 1

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.