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Given a matrix $X$. Calculating the Eigen values of $XX^T$ and using the ratio of maximum and minimum eigen values normally gives the condition number of the matrix.

If $X$ contains $M$ observations each with $N$ points. How minimizing or maximizing the ratio of eigenvalues lead to mutually independent/orthogonal observations in $X$ over multiple iterations. Considering we add an observation into $X$ if it reduces or maximizes the ratio.

Will the newly added observation in $X$ be independent from previous ones. How can it be ensured. What constraints can ensure such a thing.

Thanks.

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The question seems to presuppose that there is some connection between the computed eigenvalues, which are based on some observations of $X$, and the independence (or lack thereof) of some subsequent observation(s). There is no hint why one might expect such a connection. Indeed, there is no hint of how the subsequent observation might be influenced by the eigenvalues. After all, $X$ doesn't "know" that we've computed some eigenvalues. I'll vote to close as "not a real question". –  Andreas Blass Jan 19 '13 at 15:22
    
@Andreas The reason I assumed the connection between eigen values is the following discussion: imechanica.org/node/1915 You can also check about related concept on page 80 section 2.7 of the following book Matrix Computations: books.google.com/… A conceptual demonstration is available at: cse.illinois.edu/iem/linear_equations/condition_number If you still need explanation about a specific thing then please let me know. –  Lepy Jan 19 '13 at 17:54
    
@Lepy: Following a couple of your links, I found standard material about condition numbers, but nothing about "observations in $X$ over multiple iterations". The $X$ in your question begins as just a matrix but in the second paragraph it seems to be developing over time, with additional "observations" in anunspecified way. –  Andreas Blass Jan 21 '13 at 23:33
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