Scaling of a matrix and its impact on eigen values

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

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$b_{ij} = (1+alpha) times b_{ij}$ , when $(i-j)$ mod 2 =0

$b_{ij} = (1-alpha) times b_{ij}$ , when $(i-j)$ mod 2 =1

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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 $A$?