Undergrad Matrix Notation: ℝm x n Meaning & Vectors

[dyn]

Hi. When referring to matrices what does ℝ^{m x n} mean ? Does this notation also apply to vectors ?
Thanks

 

[fresh_42]

Hi. When referring to matrices what does ℝ^{m x n} mean ? Does this notation also apply to vectors ?
Thanks

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 $\mathbb{R}^{mn}$.

 

[haruspex]

Hi. When referring to matrices what does ℝ^{m x n} mean ? Does this notation also apply to vectors ?
Thanks

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 ℝ^mn, which would mean vectors with mn coefficients. Same dimensionality and, regarded purely as vector spaces, isomorphic, but capable of different additional structure.

 

[member 587159]

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 ℝ^mn, which would mean vectors with mn coefficients. Same dimensionality and, regarded purely as vector spaces, isomorphic, but capable of different additional structure.

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

 

[dyn]

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

 

[fresh_42]

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

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 $\mathbb{C}^2$.

 

[haruspex]

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

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

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

 

[fresh_42]

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

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 $SO(3)$?

 

[haruspex]

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?

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.

 

[fresh_42]

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 of $\mathbb{R}^{2 \times 2}\times\mathbb{R}^{2 \times 2}\rightarrow\mathbb{R}^{2 \times 2}$, but no mapping for either.

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.

 

[haruspex]

I only said it can be written as such (implying it can serve for some applications

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.

 

[dyn]

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

 

[fresh_42]

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

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.)