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This question relates to my research, and I hope that you can give me a hand. It took me more than one weeks but I could not find a satisfactory answer.

Let $X_1,X_2,..,X_k \in S^{n}_{++}$ are positive definite matrices, each matrix has size of $n \times n $. Let $I$ be the identity matrix, and let $S=X_1+X_2+..+X_k$.

We want to find an upper bound as tight as possible of

$$f=(X_1 +I)^{-1}+ (X_2 +I)^{-1}+...+(X_k +I)^{-1}$$

in terms of $S$, $k$ and $I$.

One upper bound that we can easily figure out by using the fact that $(X_i+I)^{-1} < I$. Where for two matrices $A,B$, we say $A<B$ iff $B-A$ is a positive definite matrix, i.e $B-A \in S^{n}_{++}$

Best,

Phong Le
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  • A reasonable first step would be to figure out how such a bound would look like if $X_i$ were positive scalars. – Algebraic Pavel Feb 28 '15 at 15:38
  • This was my first thought. If we think $X_i$ as positive real number, then it will be some basic inequality for high-school students.

    I do not have a good idea for now for these problems.

     – Phong Le Feb 28 '15 at 17:28

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