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Functions of matrices (Read 2045 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:
B=UAU
 
How you can do this is explained in:
Transformation of operator in another basis set  
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1201207384
 
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
Linker IT Software
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