Definition of a complex matrix

1. Sep 11, 2008

_Andreas

http://mathworld.wolfram.com/ComplexMatrix.html" [Broken] states that a complex matrix is "a matrix whose elements may contain complex numbers". My question is what the "may" means. Could a matrix be complex even if its elements does not contain any complex numbers?

Last edited by a moderator: May 3, 2017
2. Sep 11, 2008

Dick

Real matrices are contained in the set of complex matrices. In that sense they are also complex. That's the only sense in which they are complex. A real matrix is, uh, real.

3. Sep 12, 2008

_Andreas

Thanks. I have a follow-up question if you don't mind: does this mean that an element in the $$\mathbb{R}$$-vector space $$Herm_n(\mathbb{C})$$ (the set of all hermitian n x n-matrices) is also an element in $$\mathbb{C}$$^n?

Last edited: Sep 12, 2008
4. Sep 12, 2008

Dick

If you mean is the set of all hermitian matrices a subset of the set of all complex matrices, of course it is. Why do you need to ask?

5. Sep 12, 2008

_Andreas

Because I'm trying to solve a problem in which there is an expression $$AB$$, where $$A$$ is a $$\mathbb{C}$$-linear map $$A:\mathbb{C}^n\rightarrow\mathbb{C}^n$$ and $$B \in \msbox{Herm_n}(\mathbb{C})$$. If $$B$$ doesn't also lie in $$\mathbb{C}^n$$, I can't make sense of the expression.

Last edited: Sep 12, 2008
6. Sep 12, 2008

_Andreas

7. Sep 12, 2008

Hurkyl

Staff Emeritus
What's wrong with reading it as simply being the product of linear operators?

8. Sep 12, 2008

Dick

We have agreed Herm_n(C) does lie in the set nxn complex matrices, right? It's not a vector space over C because if you multiply a hermitian matrix by i it's no longer hermitian. But I don't see why that needs to concern you.

9. Sep 14, 2008

Yes.