MT 김신휘 Generalized Eigenvectors - Definition(정의)


MT Problem

Sol)

[Find eigenvalues and eigenvectors of ]

 
A = matrix(QQ,[[0,0,1,7,-1],[-5,-6,-6,-35,5],[1,1,-7,7,-1],[0,0,0,-9,0],[2,1,-5,-42,-3]])
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A = matrix(QQ,[[0,0,1,7,-1],[-5,-6,-6,-35,5],[1,1,-7,7,-1],[0,0,0,-9,0],[2,1,-5,-42,-3]])
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show(A.right_eigenvectors())
 
 



 
x1 = matrix(QQ,5,1,[0,0,0,1,7])
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x1 = matrix(QQ,5,1,[0,0,0,1,7])
 
 



[Find two linearly independent generalized eigenvectors of  belonging to ]

 
EYE = matrix(QQ,[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]])
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EYE = matrix(QQ,[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]])
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y1 = matrix(QQ,5,1,[1,0,0,0,1])
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print "rank(A+I) = " , rank(A+1*EYE)
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print "rank((A+I).augment(y1)) = " , rank((A+1*EYE).augment(y1))
 
 



 
y2 = (A+1*EYE).solve_right(y1)
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y2 = (A+1*EYE).solve_right(y1)
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print(y2)
 
 



 
z1 = matrix(QQ,5,1,[0,1,1,0,1])
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z1 = matrix(QQ,5,1,[0,1,1,0,1])
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print "rank(A+7I) = " , rank(A+7*EYE)
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print "rank((A+7I).augment(z1)) = " , rank((A+7*EYE).augment(z1))
 
 



 
z2 = (A+7*EYE).solve_right(z1)
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z2 = (A+7*EYE).solve_right(z1)
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print(z2)
 
 



 

( Citatiion : Jin Ho Kwak and Sungpyo Hong, 1997, Linear Algebra, 327-330p)


 -

 
Q = x1.augment(y1).augment(y2).augment(z1).augment(z2)
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Q = x1.augment(y1).augment(y2).augment(z1).augment(z2)
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print "Q = "
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print Q
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print
5
print "J = "
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print Q.inverse() * A * Q
 
 



 

References

Jin Ho Kwak and Sungpyo Hong, 1997, Linear Algebra, 327-330p

SKKU Matrix Theory Contents


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