What is a "secular equation"? 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!

A secular equation is another name for the characteristic equation.

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

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