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Strange result for Jordan_form()

asked 2 years ago

Cyrille gravatar image
     M = matrix([[4,-1,6],[2,1,6],[2,-1,8]])
    MM=M.jordan_form(transformation=true)
    show(MM)
    show(MM[0])
    show(MM[1])
    show(MM[1]^(-1))
    show(MM[1]^(-1)*MM[1])
    show(MM[1]^(-1)*MM[0]*MM[1])
    show(bool(MM[1]^(-1)*MM[0]*MM[1]==M))

According to the documentation and what I know of Jordan decomposition the last line should be True. No ?

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answered 2 years ago

achrzesz gravatar image

updated 2 years ago

Jordan canonical form is given by J = T^{-1} A T

A = matrix([[4,-1,6],[2,1,6],[2,-1,8]])
J, T = A.jordan_form(transformation=True)
T^(-1)*(A * T) ==  J

True

You checked:

T^(-1)*J*T == A

which is not the same

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Note that

T*J*T^(-1) == A

True
achrzesz gravatar imageachrzesz ( 2 years ago )

Ok but my formula is in the quickref-linalg.pdf . Someone should modify it.

Cyrille gravatar imageCyrille ( 2 years ago )

And I wonder what arr the matrix obtained by my way.

Cyrille gravatar imageCyrille ( 2 years ago )

The matrix P from quickref-linalg.pdf is inverse to the transformation T. Your MM[1] corresponds to P

achrzesz gravatar imageachrzesz ( 2 years ago )

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Asked: 2 years ago

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Last updated: Jan 28 '23