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Golub mentioned "secular equation" in his Matrix Computation and a slide. However I still don't get its definition. How is a secular equation defined? Thanks!

J. M. ain't a mathematician
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Tim
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1 Answers1

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A secular equation is another name for the characteristic equation.

http://en.wikipedia.org/wiki/Characteristic_polynomial#Secular_equation

The reason it is called "secular" is because it was first used in calculations relating the planetary motion:

http://en.wikipedia.org/wiki/Secular_phenomena

http://en.wikipedia.org/wiki/Secular_variation

The etymological root of "secular" is "saeculum," meaning "of an age."

Jonathan Rich
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    Why it is called secular equation? Is there also a sacred equation? – Popopo Mar 20 '13 at 12:56
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    @Popopo See my edits above. – Jonathan Rich Mar 20 '13 at 13:04
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    @steveO: secular here means one direction of change with time, as opposed to periodic. If you have $f(t)=at+b \sin t$, the $at$ is the secular part and the $b \sin t$ is the periodic part. – Ross Millikan Mar 20 '13 at 13:22
  • It's helpful. Thank you. – Popopo Mar 20 '13 at 13:29
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    An attempt to clarify "secular" vs. "sacred": As far as I know, "saeculum" can mean either "age" or "world". The former meaning leads to secular equations, the latter leads to non-sacred things. – Andreas Blass Mar 20 '13 at 13:30
  • @RossMillikan, thank you. Can you provide a link that expands upon your answer? It is difficult searching online for the term in the context you laid out. – user1717828 Mar 31 '15 at 16:46
  • I don't agree : secular equation is a very special instance of a characteristic equation for the special case of rank-k perturbations of a matrix. See for example here – Jean Marie Sep 11 '24 at 13:03