Invariants of a characteristic polynomial

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SUMMARY

The discussion identifies three invariants of a characteristic polynomial: the trace, the determinant, and a hybrid term referred to as the second invariant. The second invariant is significant because it is a coefficient of the characteristic polynomial and remains conserved under similarity transformations. Additionally, the set of eigenvalues of a matrix is established as an invariant, with any combinations of these eigenvalues also being invariant under permutations. This concept is applicable to arbitrary sized square matrices.

PREREQUISITES
  • Understanding of characteristic polynomials
  • Familiarity with matrix theory
  • Knowledge of eigenvalues and eigenvectors
  • Concept of similarity transformations in linear algebra
NEXT STEPS
  • Study the properties of characteristic polynomials in linear algebra
  • Explore the concept of eigenvalues and their significance in matrix theory
  • Learn about similarity transformations and their effects on matrix invariants
  • Investigate the implications of invariants in higher-dimensional matrices
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Mathematicians, students of linear algebra, and anyone interested in the properties of matrices and their characteristic polynomials.

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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