19
$\begingroup$

Let $E_j$ be the $j$th largest-magnitude eigenvalue of a real symmetric $N \times N$ matrix $M$. I've found that the ratio

$$\frac{|E_1|}{\sum_{j=1}^N{|E_j|}},$$

is a measure of the "rank-one-ness" of $M$. Qualitatively, the more similar the columns of $M$ are to each other, the higher the ratio. In my graduate research, this measure appears naturally for a specific class of matrices.

I'm certain that there's been prior research on the properties and usefulness of this measure for deciding how well-aligned and similar the columns of a matrix are. For example, I've seen it used as a measure of "compressibility". Still, my searches haven't turned up much.

Where can I find out more?

  • 0
    The usual approach for treating (numerical) rank is through the concept of singular values. Those are always real and nonnegative for any matrix, unlike eigenvalues which can be complex...2012-04-14
  • 0
    @J.M.: I probably meant $E_j$ to be the absolute value of the $j$th eigenvalue.2012-04-14
  • 0
    You should edit that bit of information into your question, then.2012-04-14
  • 0
    @J.M. I realised that the matrix is always symmetric as it's a correlation coefficient matrix, so the eigenvalues are real.2012-04-15
  • 0
    I see. So here, you are essentially considering the ratio of the spectral norm to the Ky Fan norm?2012-04-19
  • 2
    @J.M.: I just looked up the Ky Fan norms, and unless I'm missing something obvious, the quantity in question is the ratio of the first Ky Fan norm to the last one.2012-04-19
  • 0
    Have you find something to read on this ratio?2012-05-08
  • 0
    It would be helpful to look at the matrix related works of R. B. Bapat.2012-05-15
  • 0
    I would look into eigenvalue theorems in spectral graph theory since the adjacency matrices of graphs are always symmetric.2012-07-05
  • 0
    Are the eigenvalues ordered so that E_1 is largest? I would use $max_j |E_j|/\sqrt{E_1^2+\cdots+E_n^2}$2012-08-02
  • 0
    Qualitatively, the more similar the columns of M are to each other, the higher the ratio. What do you mean by "similar", is this some sort of metric.2012-08-15
  • 1
    @Per In [principle component analysis](http://en.wikipedia.org/wiki/Principal_component_analysis), your formula is the proportion of variation in the first principle component, if we assume the eigenvalues are ordered from largest to smallest, i.e. $E_1 > E_2 > \cdots >E_N>0$ We do this a lot in statistics to re-orient data (without scaling) such that the $i^{th}$ eigenvalues give you a measure of variation in the $i^{th}$ eigenvector axis.2012-12-16

2 Answers 2