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I am writing a program that will need the magnitude of the largest eigenvalue of a matrix $A$. This program is being designed for essentially any square matrix.

In trying to find optimizations, I thought I would start with an upper bound on the magnitude of the largest eigenvalue of a matrix, then work from there. In my attempt, I came across an upper bound on the largest eigenvalue of a positive, real matrix (source) and read elsewhere that the theorem could be furthered to real, nonnegative matrices.

With that, my question: Are there any more upper bounds on the magnitude of the largest eigenvalue that can apply to wider classes of matrices without having to solve for any eigenvalues? By wider classes, I am mainly referring to the general case of any complex matrix but if no bound exists then the general case for real matrices, real symmetric matrices, etc.

As always, I appreciate any and all help from this community!

Chloe
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Take the absolute value of each entry and then add up the columns. The maximum sum, the L1 norm of the matrix, is an upper bound for the magnitude of any eigenvalue. Also see the Gershgorin circle theorem, which identifies the maximum absolute value sum of the rows as a bound, too.

Oscar Lanzi
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    so is the largest row (absolute values) sum – Will Jagy May 16 '22 at 23:08
  • Amazing, thank you! Would you happen to have any papers / articles / university slides that states such as a theorem? Google seems to want to hide such information. – Chloe May 17 '22 at 16:46
  • @Chloe Try to look for "Equivalence" of norms, e.g. https://en.wikipedia.org/wiki/Norm_(mathematics)#Equivalence – flawr Jan 20 '25 at 13:52