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In linear algebra we get introduced to standard polynomials that are associated to matrices such as characteristic polynomials and determinants.

What are some of the standard rational functions that can be associated to matrices?

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    $\begingroup$ What's really the point of this question? Any motivation? Thanks! $\endgroup$
    – Suvrit
    Jan 13, 2016 at 9:14

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There is a standard way to associate an algebraic curve to a linear pencil of matrices: to a one dimensional family $A+Bt$ we associate $$\det(A+Bt-\lambda I)=0.$$ When $B$ has image of dimension $1$, this is of first degree in $t$; solving with respect to $t$ we obtain a rational function associated to the pair $A,B$.

EDIT. But of course this is not the only way. As another example, you can consider $2\times (d+1)$ matrices of rank at least $1$, and associate to them rational functions of degree $d$ by using the entries in the natural order as coefficients of the numerator and denominator. Some of these functions will be reducible, and have smaller degree. This correspondence leads to a useful geometric interpretation of the Grassmannian $G(2,d+1)$, see, for example, MR1931599, MR1888795, MR2196025, MR1917479.

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  • $\begingroup$ do you mean rank(B)=1? $\endgroup$
    – Turbo
    Jan 13, 2016 at 20:09
  • $\begingroup$ @Turbo: Yes, this is what I mean. $\endgroup$
    – Alexandre Eremenko
    Jan 13, 2016 at 21:25

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