I Matrix notation

1. Oct 13, 2016

dyn

Hi. When referring to matrices what does ℝm x n mean ? Does this notation also apply to vectors ?
Thanks

2. Oct 13, 2016

Staff: Mentor

It usually means $m \times n$ real matrices, i.e. matrices with $m$ rows and $n$ columns and real numbers as entries.
You may regard every single matrix as a vector as well, if you like. So, yes.
However, if one only wants to speak about vectors in an $m \cdot n$-dimensional real space, one would probably write $\mathbb{R}^{mn}$.

3. Oct 13, 2016

haruspex

I would assume it means the space of m by n matrices with real coefficients. Not sure that I have seen it before, though. The x could be to distinguish it from ℝmn, which would mean vectors with mn coefficients. Same dimensionality and, regarded purely as vector spaces, isomorphic, but capable of different additional structure.

4. Oct 14, 2016

Math_QED

Yes, it's the set of all matrices with real coefficients. It's the same as $M_{n,m}(\mathbb{R})$

5. Oct 14, 2016

dyn

Thanks. I have seen 2x2 complex matrices described by ℂ2. Does this mean ℂ2 x 2 ? Are people using this notation interchangeably or sloppily ?

6. Oct 14, 2016

Staff: Mentor

No. It means it is .... uh, wrong.

One might write $\mathbb{C}^2$ as real $2 \times 2$-matrices, i.e. elements of $\mathbb{R}^{2 \times 2}$.

Complex
$2 \times 2$-matrices are elements of $\mathbb{C}^{2 \times 2}$.
They apply to / operate on / map vectors of $\mathbb{C}^2$.

7. Oct 14, 2016

haruspex

Surely that would be wrong too. It should mean complex vectors of two dimensions.

8. Oct 14, 2016

Staff: Mentor

I can write every $\begin{bmatrix}x_1+iy_1 \\ x_2+iy_2\end{bmatrix}\in \mathbb{C}^2$ as $\begin{bmatrix}x_1 & x_2 \\ y_1 & y_2 \end{bmatrix} \in \mathbb{R^{2 \times 2}}$, can't I?
How would you start to explain that $SU(2)$ is a cover of $SO(3)$?

9. Oct 14, 2016

haruspex

You can set up a mapping between the two, but it is not an isomorphism. There is a multiplication defined for $\mathbb{C}^2 \times \mathbb{C}^2\rightarrow\mathbb{C}$, and one defined for $\mathbb{R}^{2 \times 2}\times\mathbb{R}^{2 \times 2}\rightarrow\mathbb{R}^{2 \times 2}$, but no mapping for either.

10. Oct 14, 2016

Staff: Mentor

None of which I claimed. I only said it can be written as such (implying it can serve for some applications).
I have not claimed that it is an algebra isomorpism. However, it is an isomorphism of real vector spaces! To disqualify this as wrong, is misleading, to say the least.

11. Oct 14, 2016

haruspex

I guess that's where we differ. Writing it "as such" implies isomorphism to me. Qualified with suitable wording to explain the sense in which they are being equated (i.e. describing the mapping) is fine, but that is different.

12. Oct 14, 2016

dyn

Thanks. So as a specific example if I have the Pauli spin matrices from Quantum mechanics. Are they examples of ℂ2 or ℂ2 x 2 matrices ?

13. Oct 14, 2016

Staff: Mentor

They are complex $2 \times 2 -$matrices, i.e. elements of $\mathbb{M}(2,\mathbb{C}) \cong \mathbb{C}^{2 \times 2}$ which naturally act on the vector space $\mathbb{C}^2$.
(There are four complex matrix entries, so they have to "live" in a vector space of four complex dimensions. $\mathbb{C^2}$ has only two.)