Consider $A=U \Sigma U^{T}$ as the SVD of $A$ which is a positive definite symmetric matrix. What can we say about the eigen values and eigen vectors of $A+B, $ where, $B =(VU) \Sigma (VU)^{T}$ under the following cases?
- $V$ is an orthonormal matrix.
- $V$ is a diagonal orthonormal matrix
In each case, note that $B = VAV^T$.
For case 1, there is nothing we can say outside of the usual statement regarding the eigenvalues of a sum of positive definite matrices. In particular, because $V$ is allowed to be any orthogonal matrix, it is possible to select $V$ such that $B$ becomes any orthogonal matrix whose eigenvalues are the same as $A$'s.
The most that we can say in this case (as in the general case of the sum of two symmetric matrices) is that Weyl's inequality holds.
For case 2, note that $V$ must be diagonal with entries $\pm 1$. So, if $A$ is $n \times n$, we are limited to at most $2^{n-1}$ possibilities for $B$. Without more information about $A$, there is not much that we can say in general about what the eigenvalues of $A + B$. However, the $A + B$ will necessarily be a multiple of a "pinching" of $2A$, which (as I elaborate on below in a specific case) gives us information about the eigenvectors.
For the specific case where $A$ has size $2m$ and $V = \operatorname{diag}(I,-I)$, we have $$ VAV^T = \pmatrix{I & 0\\0 & -I} \pmatrix{A_{11} & A_{12} \\ A_{21} & A_{22}} \pmatrix{I & 0 \\ 0 & -I} = \pmatrix{A_{11} & -A_{12}\\ -A_{21} & A_{22}} \\ \implies A + B = 2\pmatrix{A_{11} & 0\\ 0 & A_{22}}. $$ In other words, what we attain is a (multiple of a) "pinching" of $A$ (cf. Bhatia's Matrix Analysis). There still isn't a lot that we can say about the eigenvalues. However, we know that we can necessarily find a basis of eigenvectors where $m$ will have $0$s in the first $m$ entries, and the other $m$ will have $0$s in the last $m$ entries.