Modules with multiple operators

cjellison · Jun 4, 2005

Consider the set of 2x2 matrices which form a ring under matrix multiplication and matrix addition.

\mathbb{R}^3 is module defined over this ring.

So, we have three dimensional vectors whose elements are 2x2 matrices.

My question: Can I also define another "scalar multiplication" that is over the field of real numbers (well, I know you can)...what is such a structure called? For example, I want it to do the following:

 3 \begin{pmatrix} \begin{pmatrix} a & b\\ c & d \end{pmatrix} & \begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}\\ \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 1 & 2\\ 4 & 3 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} 3 \begin{pmatrix} a & b\\ c & d \end{pmatrix} & 3\begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}\\ 3\begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & 3\begin{pmatrix} 1 & 2\\ 4 & 3 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} \begin{pmatrix} 3a & 3b\\ 3c & 3d \end{pmatrix} & \begin{pmatrix} 3 & 3\\ 3 & 3 \end{pmatrix}\\ \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 3 & 6\\ 12 & 9 \end{pmatrix} \end{pmatrix} 

in addition to:

 \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} \begin{pmatrix} a & b\\ c & d \end{pmatrix} & \begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}\\ \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 1 & 2\\ 4 & 3 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} a & b\\ c & d \end{pmatrix} & \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}\\ \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2\\ 4 & 3 \end{pmatrix} \end{pmatrix} =\begin{pmatrix} \begin{pmatrix} a & b\\ c & d \end{pmatrix} & \begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}\\ \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 1 & 2\\ 4 & 3 \end{pmatrix} \end{pmatrix}

fresh_42 · Aug 22, 2019

cjellison said:

Consider the set of 2x2 matrices which form a ring under matrix multiplication and matrix addition.

\mathbb{R}^3 is module defined over this ring.

Not automatically, so you have to define the module operation. How does a ##2\times 2 ## matrix operate on a three dimensional vector?

So, we have three dimensional vectors whose elements are 2x2 matrices.

This is another scenario, namely the vector space ##\left( \mathbb{M}(2,\mathbb{R}) \right)^3##.

My question: Can I also define another "scalar multiplication" that is over the field of real numbers (well, I know you can)...what is such a structure called? For example, I want it to do the following:

 3 \begin{pmatrix} \begin{pmatrix} a & b\\ c & d \end{pmatrix} & \begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}\\ \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 1 & 2\\ 4 & 3 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} 3 \begin{pmatrix} a & b\\ c & d \end{pmatrix} & 3\begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}\\ 3\begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & 3\begin{pmatrix} 1 & 2\\ 4 & 3 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} \begin{pmatrix} 3a & 3b\\ 3c & 3d \end{pmatrix} & \begin{pmatrix} 3 & 3\\ 3 & 3 \end{pmatrix}\\ \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 3 & 6\\ 12 & 9 \end{pmatrix} \end{pmatrix} 

in addition to:

 \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} \begin{pmatrix} a & b\\ c & d \end{pmatrix} & \begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}\\ \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 1 & 2\\ 4 & 3 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} a & b\\ c & d \end{pmatrix} & \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}\\ \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2\\ 4 & 3 \end{pmatrix} \end{pmatrix} =\begin{pmatrix} \begin{pmatrix} a & b\\ c & d \end{pmatrix} & \begin{pmatrix} 1 & 1\\ 1 & 1 \end{pmatrix}\\ \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix} & \begin{pmatrix} 1 & 2\\ 4 & 3 \end{pmatrix} \end{pmatrix}

No problem, it is a vector space, i.e. an ##\mathbb{R}-##module.

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