Invariants of a characteristic polynomial

AI Thread Summary
The discussion centers on the concept of invariants in the context of matrices, specifically highlighting three invariants: trace, determinant, and a hybrid term. The second invariant's classification as a hybrid is questioned, prompting a clarification that it relates to the sum of the determinants of the diagonal minors of order 2. It is noted that this quantity is conserved under similarity transformations, aligning with the properties of the characteristic polynomial's coefficients. The conversation concludes by emphasizing that any combinations of eigenvalues, which are invariant under permutations, also qualify as invariants. Understanding these invariants is crucial for analyzing matrix properties across various dimensions.
quantum123
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Hi:
There are 3 invariants. The first one is a trace. The third one is a determinant. So they are invariants.
The strange thing is the 2nd one. It is a hybrid term. Why is it also an invariant?
 
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quantum123 said:
Hi:
There are 3 invariants. The first one is a trace. The third one is a determinant. So they are invariants.
The strange thing is the 2nd one. It is a hybrid term. Why is it also an invariant?

I guess we are talking matrices in 3-dim and you are referring to the sum of the determinants of the diagonal minors of order 2. What do you mean by why? Isn't it enough that they are coefficient of the characteristic polynomial?

You can also specifically prove to yourself that this quantity is conserved under a similarity transformation (as all other coefficients).
 
Its to be expected. The set of eigenvalues of a matrix is an invariant.
So, any combinations of the eigenvalues, that is invariant under permutations
will also be an invariant.

This generalises to arbitrary sized square matrices.
 
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