Given a positive definite matrix $Q\in\mathbb{R}^{n \times n}$, I want to find a diagnonal matrix $D$ such that $rank(Q-D) \leq k < n$.


I think this can be regarded as a generalization of eigenvalue problem, which is basically problem of finding a diagonal matrix $\lambda I$ such that $rank(Q-\lambda I) < n$.


Is there any theory about this problem?



About reducing the rank of a matrix by substract a diagonal matrix
Your problem can be restated as follows: To a given symmetric matxix, can
you add a diagonal matrix so that the result has eigenvalue $0$ with high
multiplicity?


This belongs to the theory which is called Additive Inverse Eigenvalue Problems. See, for example this paper, which seems to treat a very similar problem:


D.Paul Phillips, Some partial inverse eigenvalue problems: recovering diagonal entries of symmetric matrices, Linear Algebra and its Applications
Volume 380, 263-270,


however the exact statement you ask does not follow from this result,
and I suppose that your problem is unsolved.


Here is a survey of such problems:


Moody T. Chu, Inverse eigenvalue problems, SIAM Rev. 
Vol. 40, No. 1, pp. 1–39.


