this is my first post in this web site and I hope that I find an answer to my question.
I am trying to find a closed-form expression for the inverse square root of the following symmetric band matrix of dimension N*N:
$$ M= \left( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}} 2 & 0 & 1 & 0 & \cdots & 0 & \cdots\\ 0 & 3 & 0 & 1 & 0 &\cdots & 0 & \cdots\\ 1 & 0 & 3 & 0 & 1 & \cdots & \cdots\\ 0 & 1 & 0 & 3 & 0 & 1 & \cdots & \vdots\\ \vdots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots\\ \vdots & & & & & & 3 & 0\\ 0 & \cdots & & & & & 0 & 2 \end{array} \right) $$
I am sorry if the above representation is not clear. For convenience, I include the matrix elements as follows:
$ M_{1,1}=M_{N,N}=2\\ M_{i,i}=3\ \mathrm{for}\ i=2,\ldots,N-1\\ M_{i,i+2}=M_{i+2,i}=1\ \mathrm{for}\ i=1,\ldots,N-2\\ M_{i,j}=0\ \mathrm{elsewhere} $
Is there any ideas or already done work about the problem of finding the inverse square root of the matrix $M$ defined above?
Thank you.