One Step Inverse Updating

One Step Inverse Updating - File One Step Inverse Updating...

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File: One Step Inverse Updating Consider that we have an m m × matrix 1 B that we have computed the inverse (here we are assuming that it exists) 1 1 B . We could have done this inverse by the product form method: 1 1 1 2 1 m m B M M M M = " Now assume that we want to replace column i of 1 B with the column vector a . The reason for this exchange in a simplex type algorithm is to change basic variables. So let’s denote the two matrices by 1 2 1 1 2 2 , , , , , , , , , , i m m B a a a a B a a a a = = " " " " Now we also have 1 1 B such that 1 2 1 2 1 1 1 1 1 1 2 1 2 1 1 1 2 1 , , , , , , , , , , , , , , , , , , , , i m i m m m B B B a a a a e e e e B B B a a a a e e e α = = = = " " " " " " " " The if we form an M matrix such that 1 2 1 2 , , , , , , , , , , m i m M e e e e e e e α = " " " " , then 1 1 2 1 B MB = since 1 2 1 1 1 2 1 1 2 1 2 , , , , , , , , , , , , , , , m m i m MB B MB a a a a M e e e e e e e α = = = " " " " " "
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