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September 22nd, 2015, 12:45 AM   #1
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Positive definite matrix whose smallest eigenvector is prescribed

I have two twisted questions:

1) A is a N by N positive definite symmetric matrix, thus $AP=P \Omega$, the columns of P are eigenvectors; $ \Omega$ is the diagonal matrix consist of the eigenvalues. Knoeppel’s 2013 paper says the smallest eigenvector $\psi$ could be parametrically aligned to any vector $ \phi$ by $$(A-tI)\psi = \phi$$ When t equals to the smallest eigenvalue, there is no alignment; when t approaches to $-\infty$ the alignment is stronger.

Now, My goal is to align the $i$th (instead of the smallest) eigenvector $\nu$ to $ \phi$.

2) Thus I come up with a ‘solution’: construct a new positive definite matrix B whose smallest eigenvector equals to the $i$th eigenvector $\nu$ of A, then constructing B via $$B=P \Omega P^{-1}$$ wherein P consists of the newly constructed eigenvectors of B, finally $(B-tI)\psi = \phi$ would work for my purpose.

But I doubt whether it’s possible. Suppose I want the smallest eigenvector of B equals to the 2nd (smallest) eigenvector of A. I can shift $i$th eigenvector of A as the $(i-1)$th eigenvector of B, and adding a new eigenvector $\eta$ to B. Whereas, the contradiction is: the $N-1$ eigenvectors (excludes the smallest ) of A already determinate the $\eta$ since all eigenvectors are pair-wise orthogonal.

Another strategy might be keeping the $i$th eigenvector $\nu$ of A as the smallest eigenvector of B, and invent all $N-1$ eigenvectors of B. It is equivalent to find an arbitrary set of orthogonal vectors in the $N-1$ dimensional subspace. However, how to make sure that the eigenvalue of $\nu$ is smaller than the eignvaues of all newly constructed eigenvectors?

Sorry for a long question.

Last edited by whitegreen; September 22nd, 2015 at 12:51 AM.
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September 25th, 2015, 06:41 AM   #2
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The "smallest eigenvector" makes no sense because any multiple of an eigenvector is an eigenvector. I think you mean "an eigenvector corresponding to the smallest eigenvalue (in absolute value).
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