The question says that "(1,1)-entry of D is different from the other diagonal entries", but I think that in order to answer the question you need to know more than that, namely that the other two diagonal entries of D are the same, so that D is of the form . In the absence of any information at all about the matrix A, you will surely need to make that assumption about D.

If then . The idea is to write P in the form , where Q is orthogonal and R is a positive definite matrix that commutes with D. Then , from which .

The factorisation P = QR is called the polar decomposition of P. The positive definite factor R is the square root of . So start by calculating . This has square root . (Fortunately, this commutes with D, as you can easily check.) Then .

This is much easier. The diagonal entries of D are the eigenvalues of A, and the columns of Q are the associated normalised eigenvectors.