# Types of complex matrices, why only 3?

• I
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!

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Mentor
These are just three subclasses of complex matrices. They are not covering all matrices.

It is like "prime numbers" and "square numbers" for integers: Both are subsets of the integers, but they don't cover everything. In fact, most matrices (for suitable definitions of "most") are not in any of your three categories.

What do you mean by "transform"? I guess you don't want arbitrary operations, otherwise you can just exchange a matrix by the identity matrix for example.

Mentor
2022 Award
Shouldn't there be a fourth type
There are plenty. Invertible, symmetric, skew-symmetric, singular, regular, orthogonal, symplectic, special.

You are turning the process upside down. If you want to investigate a certain type of matrices and there isn't yet a name for it, then you can call this property by a suitable name. Not the other way around.

Gold Member
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!

How about the major ones : invertible, non-invertible? Tho " most" matrices are invertible, i.e., ##Gl(n, \mathbb R)## is dense in ## M_{ n\times n} \mathbb R ## , which translates to a similar result for Complex matrices (Basically, continuity of the Determinant, which is a polynomial on the entries). I assume your transformations would preserve at least this, i.e., invertibility, and you would want matrices in ##Gl(n,\mathbb C) ## to not be embeddings of those in ## Gl(n,\mathbb R) ##.

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Homework Helper
I am a novice here but I think there should indeed be, in your organization, a 4th type called hermitian symplectic. I.e. hermitian matrices are of interest for the bilinear forms they define. Among those there is a standard one defined by the identity matrix. Other hermitian matrice define this same form but with respect to a different basis, if they are positive definite. Hermitian matrices transform into other hermitian matrices by means of invertible change of basis matrices. Those matrices that preserve the standard form are called unitary.

Skew hermitian matrices also define bilinear forms, but ones that are skew (conjugate) symmetric rather than (conjugate) symmetric. There is a standard one of these also, represented by the square 2n by 2n matrix with the nbyn identity in the lower left corner, minus that matrix in the top right corner, and all else zeros. Then the fourth type of matrix would be the ones that preserve this standard skew hermitian form, the so called skew hermitian matrices.

Interestingly there is also a connection between unitary and skew hermiotian matrices. Namely it seems the lie algebra of skew hermitian matrices is naturally the tangent space at the identity of the lie group of unitary matrices! you might want to pursue some reading about bilinear forms say in Mike Artin's Algebra book, and possibly some reading on lie groups and lie algebras. The point is your chosen matrices are all associated with the theory of bilinear forms.

you might also enjoy a related but slightly different relationship arising in the theory of Riemann's theta functions. Namely these are defined by symmetric complex matrices with positive definite imaginary part, and the space of these is acted on by the real symplectic matrices, where the stabilizer of i times the identity matrix is the unitary group. This is beautifully described in chapter 2 of volume 1 of David Mumford's 3 volume treatise on theta functions.

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• WWGD
Thanks!