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Book: MultiplyingUpperTriangularMatrices


See: Math notebook

Investigation: Multiplying upper triangular matrices.


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:

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.

Parsiųstas iš http://www.ms.lt/sodas/Book/MultiplyingUpperTriangularMatrices
Puslapis paskutinį kartą pakeistas 2018 lapkričio 18 d., 19:11