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dyn
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Hi. When referring to matrices what does ℝm x n mean ? Does this notation also apply to vectors ?
Thanks
Thanks
It usually means ##m \times n## real matrices, i.e. matrices with ##m## rows and ##n## columns and real numbers as entries.dyn said:Hi. When referring to matrices what does ℝm 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 ℝmn, which would mean vectors with mn coefficients. Same dimensionality and, regarded purely as vector spaces, isomorphic, but capable of different additional structure.dyn said:Hi. When referring to matrices what does ℝm x n mean ? Does this notation also apply to vectors ?
Thanks
haruspex said: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.
No. It means it is ... uh, wrong.dyn said:Thanks. I have seen 2x2 complex matrices described by ℂ2. Does this mean ℂ2 x 2 ? Are people using this notation interchangeably or sloppily ?
Surely that would be wrong too. It should mean complex vectors of two dimensions.fresh_42 said: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}##.
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?haruspex said:Surely that would be wrong too. It should mean complex vectors of two dimensions.
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 said: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?
None of which I claimed. I only said it can be written as such (implying it can serve for some applications).haruspex said: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.
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.fresh_42 said:I only said it can be written as such (implying it can serve for some applications
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##.dyn said: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 ?
In matrix notation, "ℝm x n" refers to a matrix with m rows and n columns, where m and n are both positive integers. This notation is used to indicate the size or dimensions of a matrix.
In matrix notation, a vector is represented as a matrix with either one row or one column. For example, a column vector with three elements would be written as a 3x1 matrix, and a row vector with four elements would be written as a 1x4 matrix.
Matrix notation is used in mathematics to simplify and generalize the representation of mathematical operations involving multiple variables. It allows for concise and organized notation, making it easier to perform calculations and solve equations.
Vectors can be thought of as a special case of matrices, where one dimension is equal to 1. In fact, a 1xN matrix is equivalent to a row vector, and an Nx1 matrix is equivalent to a column vector. Matrices can also be used to perform operations on vectors, such as addition, subtraction, and multiplication.
The "ℝ" symbol in matrix notation stands for the set of real numbers. This indicates that the elements within the matrix or vector are real numbers, as opposed to complex numbers, which are represented by the symbol "ℂ".