Introduction E9F5FC

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Consider a set {$S$} of matrices {$M$} whose entries {$M_{ij}$} satisfy {$i\geq j+k$} for some fixed {$k$}. If {$k=0$}, then these are the upper triangular matrices, and if {$k=1$}, then these are the strictly upper triangular matrices.

We can interpret each entry {$M_{ij}$} as the possible paths from {$i$} to {$j$}. Multiplication of matrices is composition of paths. Thus the product of r matrices will satisfy the equation {$i\geq j+rk$}.

Suppose one matrix satisfies {$i\geq j+k$} and the other satisifes {$i\geq j+l$}. Then multiplying the two matrices together will result in a matrix satisfying {$i\geq j+k+l$}.

In particular:

• upper triangular matrices are closed under multiplication.
• strictly upper triangular matrices yield a series of ever more restrictive subalgebras of matrices.

If we consider the Lie bracket of two matrices, then it will be the difference of two products of the matrices, and so it will satisfy the conditions above.