0

The problem is formulated as follows:

Given $0\neq x \in \mathbb{R}^n$, and $k\leq n$, consider the following optimization problem $$\min_{\textrm{rank}(C)=k}x^t(I_n-C)^t(I_n-C)x$$ where $I_n$ be the unit matrix of $\mathbb{R}^{n\times n}$ and the minimum is taken over all $C \in \mathbb{R}^{n\times n}$ with rank of $k$.

I think this should be a standard problem, but I am not sure how to proceed with this. Any comment will be greatly appreciated!

Roy Han
  • 911
  • Is $k$ supposed to be $n$ here? Or, to clarify my real points of confusion, what is $I_k$ and what space are the $C$ coming from? I typically think of $I_k$ as the $k\times k$ identity matrix, but that wouldn't make sense here if $x \in \mathbb{R}^n$ for $k \neq n$. – Dan Oct 05 '13 at 00:53
  • @Dan Thank you for your comment, I have corrected my typo. – Roy Han Oct 05 '13 at 00:59
  • No problem! It's been a long week, and I thought I was missing something. :) – Dan Oct 05 '13 at 01:40

1 Answers1

0

Take $C = xx^t/\|x\|^2 + D$, where $D$ is any rank $k-1$ matrix with $Dx=0$. Then $C$ is of rank $k$ and $(I_n-C)x=0$.

p.s.
  • 6,401