Definition of a complex matrix

[_Andreas]

http://mathworld.wolfram.com/ComplexMatrix.html" 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?

 

[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.

 

[_Andreas]

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.

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?

 

[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?

 

[_Andreas]

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?

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.

 

[_Andreas]

Adding to my confusion was this statement on Wikipedia:

The Hermitian n-by-n matrices form a vector space over the real numbers (but not over the complex numbers).

 

[Hurkyl]

I can't make sense of the expression.

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

 

[Dick]

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.

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.

 

[_Andreas]

We have agreed Herm_n(C) does lie in the set nxn complex matrices, right?

Yes.

It's not a vector space over C because if you multiply a hermitian matrix by i it's no longer hermitian.

Very helpful! Thanks.

But I don't see why that needs to concern you.

I've actually solved the problem, and you're right, it was unnecessary for me to be concerned by it.

Thanks again for your help.