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Question: Eigenvectors from a symmetric matrix

Question:Eigenvectors from a symmetric matrix

Inacio Aprendanet 65 Maple
matrix computation symmetric
March 26 2013
0

I was working with the computation of the eigenvectors of a 3X3 symmetric matrix with algebraic entries and Maple 17 doesn´t give me an answer after a long time, even with CUDA activated. You can see this by the commands below:

 


 

 

 

> 

restart
> 

CUDA:-IsEnabled();

false

 (1)
> 

CUDA:-Enable()

false

 (2)
> 

CUDA:-IsEnabled();

true

 (3)
> 

M := Matrix([[m[1, 1], m[1, 2], m[1, 3]], [m[1, 2], m[2, 2], m[2, 3]], [m[1, 3], m[2, 3], m[3, 3]]])

M := Matrix(3, 3, {(1, 1) = m[1, 1], (1, 2) = m[1, 2], (1, 3) = m[1, 3], (2, 1) = m[1, 2], (2, 2) = m[2, 2], (2, 3) = m[2, 3], (3, 1) = m[1, 3], (3, 2) = m[2, 3], (3, 3) = m[3, 3]})

 (4)
> 

det := LinearAlgebra:-Determinant(M)

m[1, 1]*m[2, 2]*m[3, 3]-m[1, 1]*m[2, 3]^2-m[1, 2]^2*m[3, 3]+2*m[1, 2]*m[1, 3]*m[2, 3]-m[1, 3]^2*m[2, 2]

 (5)
> 

m[1, 2] := solve(det, m[1, 2])[1]

(m[1, 3]*m[2, 3]+(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2)^(1/2))/m[3, 3]

 (6)
> 

K := Matrix(3, 3, {(1, 1) = k[1, 1], (1, 2) = k[2, 1], (1, 3) = k[3, 1], (2, 1) = k[2, 1], (2, 2) = k[2, 2], (2, 3) = k[3, 2], (3, 1) = k[3, 1], (3, 2) = k[3, 2], (3, 3) = k[3, 3]})

K := Matrix(3, 3, {(1, 1) = k[1, 1], (1, 2) = k[2, 1], (1, 3) = k[3, 1], (2, 1) = k[2, 1], (2, 2) = k[2, 2], (2, 3) = k[3, 2], (3, 1) = k[3, 1], (3, 2) = k[3, 2], (3, 3) = k[3, 3]})

 (7)
> 

K.M

Matrix(3, 3, {(1, 1) = k[1, 1]*m[1, 1]+k[2, 1]*(m[1, 3]*m[2, 3]+sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2))/m[3, 3]+k[3, 1]*m[1, 3], (1, 2) = k[1, 1]*(m[1, 3]*m[2, 3]+sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2))/m[3, 3]+k[2, 1]*m[2, 2]+k[3, 1]*m[2, 3], (1, 3) = k[1, 1]*m[1, 3]+k[2, 1]*m[2, 3]+k[3, 1]*m[3, 3], (2, 1) = k[2, 1]*m[1, 1]+k[2, 2]*(m[1, 3]*m[2, 3]+sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2))/m[3, 3]+k[3, 2]*m[1, 3], (2, 2) = k[2, 1]*(m[1, 3]*m[2, 3]+sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2))/m[3, 3]+k[2, 2]*m[2, 2]+k[3, 2]*m[2, 3], (2, 3) = k[2, 1]*m[1, 3]+k[2, 2]*m[2, 3]+k[3, 2]*m[3, 3], (3, 1) = k[3, 1]*m[1, 1]+k[3, 2]*(m[1, 3]*m[2, 3]+sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2))/m[3, 3]+k[3, 3]*m[1, 3], (3, 2) = k[3, 1]*(m[1, 3]*m[2, 3]+sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2))/m[3, 3]+k[3, 2]*m[2, 2]+k[3, 3]*m[2, 3], (3, 3) = k[3, 1]*m[1, 3]+k[3, 2]*m[2, 3]+k[3, 3]*m[3, 3]})

 (8)
> 

> 

NULL
> 

det := LinearAlgebra:-Eigenvalues(K.M)

det := Vector(3, {(1) = 0, (2) = ((1/2)*k[1, 1]*m[1, 1]*m[3, 3]+k[2, 1]*m[1, 3]*m[2, 3]+(1/2)*k[2, 2]*m[2, 2]*m[3, 3]+k[3, 1]*m[1, 3]*m[3, 3]+k[3, 2]*m[2, 3]*m[3, 3]+(1/2)*k[3, 3]*m[3, 3]^2+sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2)*k[2, 1]+(1/2)*sqrt(-2*k[2, 2]*k[3, 3]*m[2, 2]*m[3, 3]^3+4*k[2, 2]*k[3, 3]*m[2, 3]^2*m[3, 3]^2-4*sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2)*k[2, 1]*k[3, 3]*m[3, 3]^2+8*sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2)*k[3, 1]*k[3, 2]*m[3, 3]^2+4*k[3, 1]*k[3, 3]*m[1, 3]*m[3, 3]^3+4*k[3, 2]*k[3, 3]*m[2, 3]*m[3, 3]^3+4*k[2, 1]^2*m[1, 1]*m[2, 2]*m[3, 3]^2+4*k[3, 2]^2*m[2, 2]*m[3, 3]^3+k[2, 2]^2*m[2, 2]^2*m[3, 3]^2+4*k[3, 1]^2*m[1, 1]*m[3, 3]^3+k[1, 1]^2*m[1, 1]^2*m[3, 3]^2+k[3, 3]^2*m[3, 3]^4+2*k[1, 1]*k[2, 2]*m[1, 1]*m[2, 2]*m[3, 3]^2+4*k[1, 1]*k[3, 1]*m[1, 1]*m[1, 3]*m[3, 3]^2+4*k[2, 1]*k[3, 3]*m[1, 3]*m[2, 3]*m[3, 3]^2+4*k[2, 2]*k[3, 2]*m[2, 2]*m[2, 3]*m[3, 3]^2+8*k[3, 1]*k[3, 2]*m[1, 3]*m[2, 3]*m[3, 3]^2+4*sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2)*k[1, 1]*k[2, 1]*m[1, 1]*m[3, 3]+4*sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2)*k[2, 1]*k[2, 2]*m[2, 2]*m[3, 3]+8*k[2, 1]*k[3, 2]*m[1, 3]*m[2, 2]*m[3, 3]^2-4*k[2, 2]*k[3, 1]*m[1, 3]*m[2, 2]*m[3, 3]^2+8*k[2, 2]*k[3, 1]*m[1, 3]*m[2, 3]^2*m[3, 3]+8*sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2)*k[1, 1]*k[2, 2]*m[1, 3]*m[2, 3]+8*sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2)*k[1, 1]*k[3, 2]*m[1, 3]*m[3, 3]+8*sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2)*k[2, 2]*k[3, 1]*m[2, 3]*m[3, 3]-4*k[1, 1]*k[2, 2]*m[1, 1]*m[2, 3]^2*m[3, 3]-4*k[1, 1]*k[2, 2]*m[1, 3]^2*m[2, 2]*m[3, 3]-4*k[1, 1]*k[3, 2]*m[1, 1]*m[2, 3]*m[3, 3]^2+8*k[2, 1]*k[3, 1]*m[1, 1]*m[2, 3]*m[3, 3]^2+8*k[1, 1]*k[3, 2]*m[1, 3]^2*m[2, 3]*m[3, 3]+4*k[1, 1]*k[2, 1]*m[1, 1]*m[1, 3]*m[2, 3]*m[3, 3]+8*k[1, 1]*k[2, 2]*m[1, 3]^2*m[2, 3]^2-2*k[1, 1]*k[3, 3]*m[1, 1]*m[3, 3]^3+4*k[1, 1]*k[3, 3]*m[1, 3]^2*m[3, 3]^2+4*k[2, 1]*k[2, 2]*m[1, 3]*m[2, 2]*m[2, 3]*m[3, 3]))/m[3, 3], (3) = ((1/2)*k[1, 1]*m[1, 1]*m[3, 3]+k[2, 1]*m[1, 3]*m[2, 3]+(1/2)*k[2, 2]*m[2, 2]*m[3, 3]+k[3, 1]*m[1, 3]*m[3, 3]+k[3, 2]*m[2, 3]*m[3, 3]+(1/2)*k[3, 3]*m[3, 3]^2+sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2)*k[2, 1]-(1/2)*sqrt(-2*k[2, 2]*k[3, 3]*m[2, 2]*m[3, 3]^3+4*k[2, 2]*k[3, 3]*m[2, 3]^2*m[3, 3]^2-4*sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2)*k[2, 1]*k[3, 3]*m[3, 3]^2+8*sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2)*k[3, 1]*k[3, 2]*m[3, 3]^2+4*k[3, 1]*k[3, 3]*m[1, 3]*m[3, 3]^3+4*k[3, 2]*k[3, 3]*m[2, 3]*m[3, 3]^3+4*k[2, 1]^2*m[1, 1]*m[2, 2]*m[3, 3]^2+4*k[3, 2]^2*m[2, 2]*m[3, 3]^3+k[2, 2]^2*m[2, 2]^2*m[3, 3]^2+4*k[3, 1]^2*m[1, 1]*m[3, 3]^3+k[1, 1]^2*m[1, 1]^2*m[3, 3]^2+k[3, 3]^2*m[3, 3]^4+2*k[1, 1]*k[2, 2]*m[1, 1]*m[2, 2]*m[3, 3]^2+4*k[1, 1]*k[3, 1]*m[1, 1]*m[1, 3]*m[3, 3]^2+4*k[2, 1]*k[3, 3]*m[1, 3]*m[2, 3]*m[3, 3]^2+4*k[2, 2]*k[3, 2]*m[2, 2]*m[2, 3]*m[3, 3]^2+8*k[3, 1]*k[3, 2]*m[1, 3]*m[2, 3]*m[3, 3]^2+4*sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2)*k[1, 1]*k[2, 1]*m[1, 1]*m[3, 3]+4*sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2)*k[2, 1]*k[2, 2]*m[2, 2]*m[3, 3]+8*k[2, 1]*k[3, 2]*m[1, 3]*m[2, 2]*m[3, 3]^2-4*k[2, 2]*k[3, 1]*m[1, 3]*m[2, 2]*m[3, 3]^2+8*k[2, 2]*k[3, 1]*m[1, 3]*m[2, 3]^2*m[3, 3]+8*sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2)*k[1, 1]*k[2, 2]*m[1, 3]*m[2, 3]+8*sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2)*k[1, 1]*k[3, 2]*m[1, 3]*m[3, 3]+8*sqrt(m[1, 1]*m[2, 2]*m[3, 3]^2-m[1, 1]*m[2, 3]^2*m[3, 3]-m[1, 3]^2*m[2, 2]*m[3, 3]+m[1, 3]^2*m[2, 3]^2)*k[2, 2]*k[3, 1]*m[2, 3]*m[3, 3]-4*k[1, 1]*k[2, 2]*m[1, 1]*m[2, 3]^2*m[3, 3]-4*k[1, 1]*k[2, 2]*m[1, 3]^2*m[2, 2]*m[3, 3]-4*k[1, 1]*k[3, 2]*m[1, 1]*m[2, 3]*m[3, 3]^2+8*k[2, 1]*k[3, 1]*m[1, 1]*m[2, 3]*m[3, 3]^2+8*k[1, 1]*k[3, 2]*m[1, 3]^2*m[2, 3]*m[3, 3]+4*k[1, 1]*k[2, 1]*m[1, 1]*m[1, 3]*m[2, 3]*m[3, 3]+8*k[1, 1]*k[2, 2]*m[1, 3]^2*m[2, 3]^2-2*k[1, 1]*k[3, 3]*m[1, 1]*m[3, 3]^3+4*k[1, 1]*k[3, 3]*m[1, 3]^2*m[3, 3]^2+4*k[2, 1]*k[2, 2]*m[1, 3]*m[2, 2]*m[2, 3]*m[3, 3]))/m[3, 3]})

 (9)
> 

det := LinearAlgebra:-Eigenvectors(Typesetting:-delayDotProduct(K, M))

Warning,  computation interrupted

  
> 

``


How can I perform the calculation? Is this a problem from my computer or the commands that I was using?

 

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