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Matrix diagonalisation (Read 1831 times)
Gerrit-Jan Linker
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Matrix diagonalisation
25.09.13 at 08:54:52
 
Matrix diagonalisation
 
To diagonalise a matrix A one needs to find another matrix C to bring A in its diagonal form D. One also needs the inverse of matrix C, C-1:
D=C-1AC
 
To find matrix C the eigenvalues and eigenvectors of A need to be computed first. This can be done using the characteristic polynomial F:
F(x)=det(A-xI)=0 where I is the unit matrix and det is short for determinant.
 
We continue by example. Let  

A =  2  6
     0  -1
 
F(x) =  det( 2-x  6
             0  -1-x)
= (2-x)(-1-x)-6.0 = (2-x)(-1-x) =0

Hence the eigenvalues are x=2 and x=-1.
 
To find the corresponding eigenvectors, substitute the eigenvalues back into A-xI=0.
The two eigenvectors are (1,0) and (-2,1). Arranging these into a matrix gives matrix C:

C =  1  -2
     0  1
 
Its inverse is  
C-1 =  1  2
       0  1

 
Now we can calculate the diagonal matrix D from: D=C-1AC

D =  (1  2)  (2   6) (1 -2)
     (0  1)  (0  -1) (0  1)
D =  (1  2)  (2   2)  
     (0  1)  (0  -1)  
D =  (2  0)  
     (0  -1)  

 
 
See also:
https://www.math.okstate.edu/~binegar/3013-S99/3013-l16.pdf
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« Last Edit: 25.09.13 at 09:15:25 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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