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There are $n$ symmetric polynomials in the eigenvalues of a square matrix. Two of these are the determinant and the trace, each of which have countless applications and interpretations in algebra and geometry.

What about the other symmetric polynomials? They are also similarity invariants, yet I've never seen them used or referenced. Are there any geometric interpretations, or applications, for these other invariants?

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Of course, the invariant that collects all those together is the characteristic polynomial. –  lhf Feb 26 '11 at 19:50

2 Answers 2

up vote 9 down vote accepted

The trace and the determinant are the most useful invariants because the trace is additive and the determinant is multiplicative. The other coefficients of the characteristic polynomial are neither. The determinant also has a clear geometric interpretation. In addition, all of the coefficients of the characteristic polynomial of an operator $T$ can be computed from the traces of the operators $T^n$; this is one reason why it is not so surprising that traces of group elements in group representations carry a lot of information.

This is not to say that people never use the other invariants, although they don't tend to have special names. For example, my understanding is that the Killing form in Lie theory, an important tool, was discovered by messing around with characteristic polynomials. And the construction underlying the coefficients of the characteristic polynomial, the exterior algebra, is enormously useful.

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I think you are thinking of the coefficients of the characteristic polynomial -the trace and the determinant are the "extremes". See eg http://alert.comule.com/?p=7741 and http://mathoverflow.net/questions/33478/geometric-interpretation-of-characteristic-polynomial

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I think the question can be seen as a duplicate then. –  lhf Feb 26 '11 at 20:01

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