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Functions of matrices (Read 2875 times)
Gerrit-Jan Linker
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Functions of matrices
26.01.08 at 11:20:55
Functions of matrices
Diagonalisation of matrices can be used when calculating functions of matrices f(A). If the matrix A is diagonal f(A) can be calculated as f(A) = f(Aii) for all values of i.
In general when we have a matrix A it is not diagonal. The proces of obtaining f(A) is to diagonalise A to give the diagonal matrix B. We can compute f(B) by computing f(Bii) for all the diagonal elements (all i)
To obtain the diagonal matrix B of matrix A you need to find the diagonalisation matrix U:
How you can do this is explained in:
Transformation of operator in another basis set
After computing the function of the diagonal elements of the matrix the function can be applied to the diagonal matrix elements f(Bii). We get a new matrix, say C where Cii = f(Bii) on the diagonal.  
To give the answer for f(A) we just need to reverse the diagonalisation. Suppose D is the resulting matrix of f(A). We use the following to obtain D.
f(A) = D = UCU
Another way to write this is using f(B) where B is the diagonal of A:
f(A) = D = Uf(B)U
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« Last Edit: 26.01.08 at 11:42:14 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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