Given a positive definite symmetric matrix $A$ of size $n \times n$ where $n$ is even, with known eigen values and eigen vectors, what can we say about the eigen values/vectors of a new matrix $B$ formed as
$b_{ij} = (1+\alpha) \times b_{ij}$ , when $(i-j)$ mod 2 =0
$b_{ij} = (1-\alpha) \times b_{ij}$ , when $(i-j)$ mod 2 =1
In effect, we are scaling all the even diagonals of $A$ by $1+\alpha$ and all the odd diagonals by $1-\alpha$.
NB: $\alpha$ is a real positive scalar.
Also , I understand that when $\alpha =1$ ,as shown here, we could transform the matrix B to a block diagonal matrix and could find the eigen values of $B$ easily. But my question is what happens when $\alpha$ is a generic scalar.
Also, If there does not exist any direct relation between eigen values of $A$ and $B$ ,is there any approximate function which I could use to approximate eigen values of $B$ using those of $A$?