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Hi, the three main types of complex matrices are:

1. Hermitian, with only real eigenvalues

2. Skew-Hermitian , with only imaginary eigenvalues

3. Unitary, with only complex conjugates.

Shouldn't there be a fourth type:

4. Non-unitary-non-hermitian, with one imaginary value (i.e. 3i) and a complex conjugate (i.e. 4+5i)

?

Can such a forth type be transformed in some fashion? It is not diagonalizable, not normal and not-Hermitian. I have looked into the Foldy–Wouthuysen transformation , but am not sure this will work on a complex matrix. Are there any possibilities to transform a matrix of class 4 to one of class 1, such as using Gauss Elimination?

Thanks!

1. Hermitian, with only real eigenvalues

2. Skew-Hermitian , with only imaginary eigenvalues

3. Unitary, with only complex conjugates.

Shouldn't there be a fourth type:

4. Non-unitary-non-hermitian, with one imaginary value (i.e. 3i) and a complex conjugate (i.e. 4+5i)

?

Can such a forth type be transformed in some fashion? It is not diagonalizable, not normal and not-Hermitian. I have looked into the Foldy–Wouthuysen transformation , but am not sure this will work on a complex matrix. Are there any possibilities to transform a matrix of class 4 to one of class 1, such as using Gauss Elimination?

Thanks!

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