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Question: Implicit=false in Eigenvectors() vs Eigenvalues()

Question:Implicit=false in Eigenvectors() vs Eigenvalues()

itsme 769 Maple
linearalgebra eigenvalues
April 24 2019
1

how can i get an 'implicit=false' form (no RootOf) of my eigenvalues from the Eigenvectors() command? Should one expect the behavior be the same in Eigenvectors() and Eigenvalues()?

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M:=Matrix(3, 3, [[-2*lambda-kappa__c, -2*sqrt(2)*g, 0], [(1/2)*g*N*sqrt(2), -lambda-(1/2)*kappa__c-gamma__phi, -sqrt(2)*g], [0, g*N*sqrt(2), -2*gamma__phi]])

Matrix(3, 3, {(1, 1) = -2*lambda-`#msub(mi("κ",fontstyle = "normal"),mi("c"))`, (1, 2) = -2*sqrt(2)*g, (1, 3) = 0, (2, 1) = (1/2)*g*N*sqrt(2), (2, 2) = -lambda-(1/2)*`#msub(mi("κ",fontstyle = "normal"),mi("c"))`-`#msub(mi("γ",fontstyle = "normal"),mi("φ",fontstyle = "normal"))`, (2, 3) = -sqrt(2)*g, (3, 1) = 0, (3, 2) = g*N*sqrt(2), (3, 3) = -2*`#msub(mi("γ",fontstyle = "normal"),mi("φ",fontstyle = "normal"))`})

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LinearAlgebra:-Eigenvalues(M); #this is what i want but from Eigenvectors()

Vector(3, {(1) = -lambda-(1/2)*`#msub(mi("κ",fontstyle = "normal"),mi("c"))`-`#msub(mi("γ",fontstyle = "normal"),mi("φ",fontstyle = "normal"))`, (2) = -lambda-(1/2)*`#msub(mi("κ",fontstyle = "normal"),mi("c"))`-`#msub(mi("γ",fontstyle = "normal"),mi("φ",fontstyle = "normal"))`+(1/2)*sqrt(-16*N*g^2+4*lambda^2+4*lambda*`#msub(mi("κ",fontstyle = "normal"),mi("c"))`-8*lambda*`#msub(mi("γ",fontstyle = "normal"),mi("φ",fontstyle = "normal"))`+`#msub(mi("κ",fontstyle = "normal"),mi("c"))`^2-4*`#msub(mi("κ",fontstyle = "normal"),mi("c"))`*`#msub(mi("γ",fontstyle = "normal"),mi("φ",fontstyle = "normal"))`+4*`#msub(mi("γ",fontstyle = "normal"),mi("φ",fontstyle = "normal"))`^2), (3) = -lambda-(1/2)*`#msub(mi("κ",fontstyle = "normal"),mi("c"))`-`#msub(mi("γ",fontstyle = "normal"),mi("φ",fontstyle = "normal"))`-(1/2)*sqrt(-16*N*g^2+4*lambda^2+4*lambda*`#msub(mi("κ",fontstyle = "normal"),mi("c"))`-8*lambda*`#msub(mi("γ",fontstyle = "normal"),mi("φ",fontstyle = "normal"))`+`#msub(mi("κ",fontstyle = "normal"),mi("c"))`^2-4*`#msub(mi("κ",fontstyle = "normal"),mi("c"))`*`#msub(mi("γ",fontstyle = "normal"),mi("φ",fontstyle = "normal"))`+4*`#msub(mi("γ",fontstyle = "normal"),mi("φ",fontstyle = "normal"))`^2)})

 (2)
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LinearAlgebra:-Eigenvalues(M, implicit=true); #RootOf notation; not what i want, but option works here

Vector(3, {(1) = -lambda-(1/2)*`#msub(mi("κ",fontstyle = "normal"),mi("c"))`-`#msub(mi("γ",fontstyle = "normal"),mi("φ",fontstyle = "normal"))`, (2) = RootOf(_Z^2+(2*lambda+`#msub(mi("κ",fontstyle = "normal"),mi("c"))`+2*`#msub(mi("γ",fontstyle = "normal"),mi("φ",fontstyle = "normal"))`)*_Z+4*N*g^2+4*lambda*`#msub(mi("γ",fontstyle = "normal"),mi("φ",fontstyle = "normal"))`+2*`#msub(mi("γ",fontstyle = "normal"),mi("φ",fontstyle = "normal"))`*`#msub(mi("κ",fontstyle = "normal"),mi("c"))`, index = 1), (3) = RootOf(_Z^2+(2*lambda+`#msub(mi("κ",fontstyle = "normal"),mi("c"))`+2*`#msub(mi("γ",fontstyle = "normal"),mi("φ",fontstyle = "normal"))`)*_Z+4*N*g^2+4*lambda*`#msub(mi("γ",fontstyle = "normal"),mi("φ",fontstyle = "normal"))`+2*`#msub(mi("γ",fontstyle = "normal"),mi("φ",fontstyle = "normal"))`*`#msub(mi("κ",fontstyle = "normal"),mi("c"))`, index = 2)})

 (3)
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evals, evecs:=LinearAlgebra:-Eigenvectors(M); #RootOf everywhere;

evals, evecs := Vector(3, {(1) = RootOf(_Z^2+(2*lambda+`κ__c`+2*`γ__φ`)*_Z+4*g^2*N+4*`γ__φ`*lambda+2*`γ__φ`*`κ__c`, index = 1), (2) = RootOf(_Z^2+(2*lambda+`κ__c`+2*`γ__φ`)*_Z+4*g^2*N+4*`γ__φ`*lambda+2*`γ__φ`*`κ__c`, index = 2), (3) = -lambda-(1/2)*`κ__c`-`γ__φ`}), Matrix(3, 3, {(1, 1) = 8*g^2/(RootOf(_Z^2+(2*lambda+`κ__c`+2*`γ__φ`)*_Z+4*g^2*N+4*`γ__φ`*lambda+2*`γ__φ`*`κ__c`, index = 1)+2*lambda+`κ__c`)^2, (1, 2) = 8*g^2/(RootOf(_Z^2+(2*lambda+`κ__c`+2*`γ__φ`)*_Z+4*g^2*N+4*`γ__φ`*lambda+2*`γ__φ`*`κ__c`, index = 2)+2*lambda+`κ__c`)^2, (1, 3) = 2/N, (2, 1) = -2*sqrt(2)*g/(RootOf(_Z^2+(2*lambda+`κ__c`+2*`γ__φ`)*_Z+4*g^2*N+4*`γ__φ`*lambda+2*`γ__φ`*`κ__c`, index = 1)+2*lambda+`κ__c`), (2, 2) = -2*sqrt(2)*g/(RootOf(_Z^2+(2*lambda+`κ__c`+2*`γ__φ`)*_Z+4*g^2*N+4*`γ__φ`*lambda+2*`γ__φ`*`κ__c`, index = 2)+2*lambda+`κ__c`), (2, 3) = -(1/4)*sqrt(2)*(2*lambda+`κ__c`-2*`γ__φ`)/(g*N), (3, 1) = 1, (3, 2) = 1, (3, 3) = 1})

 (4)
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evals, evecs:=LinearAlgebra:-Eigenvectors(M, implicit=false); #still RootOf everywhere;

evals, evecs := Vector(3, {(1) = RootOf(_Z^2+(2*lambda+`κ__c`+2*`γ__φ`)*_Z+4*g^2*N+4*`γ__φ`*lambda+2*`γ__φ`*`κ__c`, index = 1), (2) = RootOf(_Z^2+(2*lambda+`κ__c`+2*`γ__φ`)*_Z+4*g^2*N+4*`γ__φ`*lambda+2*`γ__φ`*`κ__c`, index = 2), (3) = -lambda-(1/2)*`κ__c`-`γ__φ`}), Matrix(3, 3, {(1, 1) = 8*g^2/(RootOf(_Z^2+(2*lambda+`κ__c`+2*`γ__φ`)*_Z+4*g^2*N+4*`γ__φ`*lambda+2*`γ__φ`*`κ__c`, index = 1)+2*lambda+`κ__c`)^2, (1, 2) = 8*g^2/(RootOf(_Z^2+(2*lambda+`κ__c`+2*`γ__φ`)*_Z+4*g^2*N+4*`γ__φ`*lambda+2*`γ__φ`*`κ__c`, index = 2)+2*lambda+`κ__c`)^2, (1, 3) = 2/N, (2, 1) = -2*sqrt(2)*g/(RootOf(_Z^2+(2*lambda+`κ__c`+2*`γ__φ`)*_Z+4*g^2*N+4*`γ__φ`*lambda+2*`γ__φ`*`κ__c`, index = 1)+2*lambda+`κ__c`), (2, 2) = -2*sqrt(2)*g/(RootOf(_Z^2+(2*lambda+`κ__c`+2*`γ__φ`)*_Z+4*g^2*N+4*`γ__φ`*lambda+2*`γ__φ`*`κ__c`, index = 2)+2*lambda+`κ__c`), (2, 3) = -(1/4)*sqrt(2)*(2*lambda+`κ__c`-2*`γ__φ`)/(g*N), (3, 1) = 1, (3, 2) = 1, (3, 3) = 1})

 (5)
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