# Homework Help: 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.