Undergrad Can a Hermitian matrix have complex eigenvalues?

[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

 

[fresh_42]

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 ...?

 

[WWGD]

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...

 

[SeM]

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.

 

[SeM]

Please have a look at this excellent reference to further classify complex matrices.

http://math.nsc.ru/LBRT/u2/Survey paper.pdf