# generalizations of determinant and trace

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

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.