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My Question

What methods can I use to estimate (or find an upper bound) the largest eigenvalue of a symmetric matrix? What if my matrix has semi-positive (zero or positive) off diagonal entries?

  • Note I am not asking how to numerically calculate the largest eigenvalue. I am trying to derive some analytical results.
  • Any references would be greatly appreciated.
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The following paper is quite old, but lists several theorems on eigenvalue bounds. Maybe it's a useful starting point: https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19680007865.pdf

Notably, for semi-positive matrices the Perron-Frobenius theorem (or alternatively the Gershgorin Circle Theorem) tell you that the largest eigenvalue lies between the minimum and maximum row sum. But maybe there are tighter bounds if you know something special about your matrix.

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  • $\begingroup$ Unfortunately the diagonals of my matrix are sometimes negative so Peron-frobenius theorem doesn't apply. $\endgroup$ – AzJ May 29 '19 at 19:15

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