How to find which particular eigenvalue will increase by $1$? I have a symmetric matrix.
A=\begin{bmatrix}2&2&2&2&2&2\\2&4&4&4&4&4\\2&4&6&6&6&6\\2&4&6&8&8&8\\2&4&6&8&10&10\\2&4&6&8&10&12\\\end{bmatrix} and 
B=\begin{bmatrix}2&0&0&0&0&0\\0&2&0&0&0&0\\0&0&2&0&0&0\\0&0&0&2&0&0\\0&0&0&0&2&0\\0&0&0&0&0&1\\\end{bmatrix} If I know the spectrum of $A$, Can I predict spectrum of $A+B$ ?
EDIT: Now I realize that matrices don't commute here. So how to proceed in this case?
[ Both matrices are real symmetric and commuting, eigenvalues are going to be added in $A+B$ 
i.e. If $\{\lambda_1,\lambda_2,...\lambda_6\}$ are ev of $A$ then in spectrum of $A+B$ these values of $\lambda_i$ increase by $2$ except one which increase by $1$. How to find which particular eigenvalue will increase by $1$? Any reference?] 
 A: Maybe will help a bit on the way.
$B = I_6 + (I_5\oplus 0)$, so adding with $I_6$ should increase each eigenvalue by 1 (regardless of eigenvectors of $A$). Then depending on the algebraic structure of $A$, adding $I_5\oplus 0$ could do different things, but we know it will distribute the remaining $5$ from $trace(I_5\oplus 0)$ among all the eigenvalues. But since $A$ and $B$ don't commute adding will also change the eigenvectors. So we can't calculate the change in terms of $A$ eigenvectors as they won't be the same as $A+B$ eigenvectors.
However if we allow ourselves to look at some specifics of how A looks in this case maybe we will get further. For instance seeing that it is symmetric will tell us it must have an ON-basis of eigenvectors with real non-negative eigenvalues. So the $I_6$ addition will just add to the "radius" of the ellipse. Then the $(I_5\oplus 0)$ addition will add to the radiuses in the first 5 dimensions and since the sum $(A+B)$ is also symmetric change the eigensystem so it becomes a new ellipse with increased width in the prescribed dimensions.
