The basis of n x n matrices with matrix multiplication!

  • Thread starter etnad179
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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 [tex] A,B \in M[/tex] then [tex] C_{ij} = (A \times B )_{ij} = \sum_k A_{ik} B_{kj} [/tex]

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 [tex] A_{i0} \& B_{0j} [/tex] for [tex] i,j=0,...,n [/tex] 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!
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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. )

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