Can a Hermitian matrix have complex eigenvalues?

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Discussion Overview

The discussion revolves around the classification of a matrix that has complex eigenvalues and complex entries, particularly in relation to its properties such as being Hermitian or non-Hermitian. Participants explore the implications of these characteristics and seek to understand how to categorize such matrices.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions whether a matrix with complex eigenvalues and complex entries can be classified as non-Hermitian.
  • Another participant asserts that all square matrices have the property of having the same determinant whether transposed or not, but acknowledges that this does not imply anything about the eigenvalues being real.
  • It is noted that a complex determinant can only arise from a matrix with complex entries, and that matrices with real entries will have real determinants.
  • Participants discuss the difficulty in classifying a complex matrix without additional properties, suggesting that more specific classifications depend on symmetry or other characteristics.
  • A reference is provided for further reading on the classification of complex matrices.

Areas of Agreement / Disagreement

Participants generally agree that the matrix in question is non-Hermitian due to its complex eigenvalues. However, there is no consensus on a more specific classification without additional properties being defined.

Contextual Notes

Participants express limitations in classification due to the lack of information about the matrix's symmetry properties and other characteristics.

SeM
Hi, I have a matrix which gives the same determinant wether it is transposed or not, however, its eigenvalues have complex roots, and there are complex numbers in the matrix elements. Can this matrix be classified as non-Hermitian?

If so, is there any other name to classify it, as it is not unitary, norm or skew-hermitian?

Thanks
 
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SeM said:
Hi, I have a matrix which gives the same determinant wether it is transposed or not, ...
All square matrices have this property, because the determinant is a symmetric polynomial in the matrix entries..
... however, its eigenvalues have complex roots, and there are complex numbers in the matrix elements. Can this matrix be classified as non-Hermitian?
Yes.
If so, is there any other name to classify it, as it is not unitary, norm or skew-hermitian?

Thanks
A complex matrix. Without any further properties known, preferably symmetry properties, it is impossible to give a more detailed answer than this, because you basically asked: How would you name a complex matrix, which is neither ...?
 
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Note that the only way you can get a Complex determinant is if you have Complex entries; manipulation of Reals leading to determinant will necessarily produce Real values, i.e., matrix with Real entries will necessarily have Real determinant, tho not necessarily Real eigenvalues nor Real n-ples of eigenvectors...
 
fresh_42 said:
All square matrices have this property, because the determinant is a symmetric polynomial in the matrix entries..

Yes.

A complex matrix. Without any further properties known, preferably symmetry properties, it is impossible to give a more detailed answer than this, because you basically asked: How would you name a complex matrix, which is neither ...?

Thanks! Very clear and concise answer.
 

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