Declare a symbolic variable with a given dimension, Then enforce non-commutative matrix algebra

I would like to declare a symbolic variable of a specific dimension/shape. for example this is 3x1 position vector, but i will never explicity write out the components of this vector in a specific coordinate system. Then define a matrix . With these defined, i would like to be able to work with expressions like and have it enforce the non-communitave matrix algebra . For example, it would throw an error if i tried or

Has anybody seen a product like this or extensions of the symbolic math toolbox to support this?

I do lots of symbolic matrix math, but almost never work with the components of the matricies.

It would be really great if we could declare a set with properties. For example the "special orthogonal group" 3 "O3" is the set of 3x3 orthogonal rotation matricies. If we could declare sets like this

then we could work with coordinate transfroms too. for example . let be a coordinate transformation from the a coordinate system to the b coordinate system.

Then i would like to declare . Now, i would like the tool to know that .