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Sturmian Separation Theorem


Let A_r=a_(ij) be a sequence of N symmetric matrices of increasing order with i,j=1, 2, ..., r and r=1, 2, ..., N. Let lambda_k(A_r) be the kth eigenvalue of A_r for k=1, 2, ..., r, where the ordering is given by

 lambda_1(A_r)>=lambda_2(A_r)>=...>=lambda_r(A_r).

Then it follows that

 lambda_(k+1)(A_(i+1))<=lambda_k(A_i)<=lambda_k(A_(i+1)).

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References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1121, 2000.

Referenced on Wolfram|Alpha

Sturmian Separation Theorem

Cite this as:

Weisstein, Eric W. "Sturmian Separation Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SturmianSeparationTheorem.html

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