The basis of n x n matrices with matrix multiplication

[etnad179]

Hi All,

I recently came across the interesting notion of constructing the minimal set of nxn matrices that can be used as a basis to generate all nxn matrices given that matrix multiplication, and addition and multiplication by scalar are allowed.

Is there a way to construct an explicit set of matrices that do this?

I'm stuck at the moment with the following thought process:

say $$A,B \in M$$ then $$C_{ij} = (A \times B )_{ij} = \sum_k A_{ik} B_{kj}$$

Now since we only need one of the products in the sum k to be non-zero - we can pick the k to be some unique value say k=0. We now have the 2n matrices $$A_{i0} \& B_{0j}$$ for $$i,j=0,...,n$$ that generate general matrices C (with a 1 in the A_{i0} or B_{0j} component since can use scalar mult.)--- But I haven't used the addition properties, so this 2n is too large (?). There is one repeat 00 so is the total 2n-1?

This would be reasonable except in the book Lie Groups, Lie Algebras... by Gilmore it says (without reason) 2(n-1), and he uses it a few times - so I presume not a typo...

Thanks in advance!

 

[fresh_42]

To generate all $n\times n$ matrices, we need $n^2$ many of them. The easiest ones are $E_{ij}$ with a one at position $(i,j)$ and zero elsewhere. Now we can write each of them as $E_{ij} = e_i \otimes e_j$ where $e_{k}$ is a vector with a one at position $k$ and zeroes elsewhere.

The tensor product here is the matrix multiplication of a column vector with a row vector.
(see e.g. https://www.physicsforums.com/insights/what-is-a-tensor/ )