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What's the best way to get the eigenvector associated with a certain eigenvalue?

Specifically, given the nature of my matrix A, I know that there ALWAYS exists an eigenvector with eigenvalue 1.  Is there a quick way to extract this without looping through the output of Eigenvectors() and checking each one?

Hi. I'm hacing trouble writing a maple procedure for the question below, can anyone help?

 

Write a maple procedure which takes as its input the vectoeat u1 and u2 and the eigenvectors lambda1 and lambda2 where u1,u2 are element of R^2 and the lambdas are real numbers.

If u1,U2 is linearly independent then the output is the matrix A an element of R^2x2 with the property that Au1= lambda1u1 and AU2=lambda2u2;

if u1,u2 is linearly dependent then the output is the statement "not an eigenbasis".

 

I I then have two inputs which I have to do but I'm not sure on how to write the procedure. Any help will be much appreciated.  

 

Thanks :)

 

 

I am having difficulty helping someone series expand an eigenvector solution.  I can expand the eigenvalues easily but get a numeric exception divide by zero when I attempt to expand a component of an eigenvector.  Mathematica seems to have no problem solving this problem.  Any help would be appreciated.

 

 

 

 

 

assume(varepsilon > 0)

H := Matrix(3, 3, {(1, 1) = 0, (1, 2) = -epsilon, (1, 3) = epsilon, (2, 1) = -epsilon, (2, 2) = 2-2*epsilon, (2, 3) = 0, (3, 1) = epsilon, (3, 2) = 0, (3, 3) = 2+2*epsilon})

Matrix(%id = 18446744078100429630)

(1)

with(LinearAlgebra):

evals, evecs := Eigenvectors(H):

e1 := convert(simplify(series(evals[1], varepsilon = 0, 4)), polynom)

2+2*varepsilon+(1/2)*varepsilon^2-(7/16)*varepsilon^3

(2)

e2 := convert(simplify(series(evals[2], varepsilon = 0, 4)), polynom)

-varepsilon^2

(3)

e3 := convert(simplify(series(evals[3], varepsilon = 0, 4)), polynom)

2-2*varepsilon+(1/2)*varepsilon^2+(7/16)*varepsilon^3

(4)

simplify(series(evecs[1][1], epsilon = 0, 4))

Error, (in simplify/sqrt/local) numeric exception: division by zero

 

``

 

Download CourseraOpticsEigenvalues.mwCourseraOpticsEigenvalues.mw

hi .how i can calculate eigenvector associated with the eigenvalue of the matrix.for example according attached file below

what are  eigenvector associated with the eigenvalue of matrix q which  determined as (2646.408147, 3142.030259, 6621.757707) respectively??

thanks...

eign.mw 

when one of element in matrix s variable below code is very slow

 

MA := MatrixMatrixMultiply(InputMatrix3aa - lambda*IdentityMatrix(3);
eignvalues1 := evalf(solve(Determinant(MA), lambda));
MA1 := MatrixMatrixMultiply(InputMatrix3aa - eignvalues1[1]*IdentityMatrix(3);
MA2 := MatrixMatrixMultiply(InputMatrix3aa - eignvalues1[2]*IdentityMatrix(3);
MA3 := MatrixMatrixMultiply(InputMatrix3aa - eignvalues1[3]*IdentityMatrix(3);
eignvector1 := LinearSolve(MA1,<x,y,z>);

eignvector2 := LinearSolve(MA2,<x,y,z>);

eignvector3 := LinearSolve(MA3,<x,y,z>);

daer all

how can i calculate eigenvalue and eigenvectors for a big Matrix?

example Matrix 12*12.

 

thank you

if not using

EigenvectorSol := simplify(solve({seq(seq((NewMatrix3 . NewInput3(1..-1,i))[j]=(v[i]* NewInput3(1..-1,i))[j], j=1..3), i=1..3)}, {seq(x||i, i=1..9)}));

and if want to see clearly the steps about how to solve for eignvector, how to do?

because i use solve, it has error

InputMatrix3 := Matrix([[31.25,30.8,30.5],[30.8,30.5,0],[30.5,0,0]]);

NewInput3 := MatrixMatrixMultiply(Transpose(InputMatrix3), InputMatrix3);

FirstEigenValue := solve(Determinant(NewInput3-Matrix([[lambda1, 0, 0], [0, lambda1, 0], [0, 0, lambda1]])), lambda1)[1]; # find back eigenvalue from eigenvector

SecondEigenValue := solve(Determinant(NewInput3-Matrix([[lambda1, 0, 0], [0, lambda1, 0], [0, 0, lambda1]])), lambda1)[2]; # find back eigenvalue from eigenvector

ThirdEigenValue := solve(Determinant(NewInput3-Matrix([[lambda1, 0, 0], [0, lambda1, 0], [0, 0, lambda1]])), lambda1)[3]; # find back eigenvalue from eigenvector

v:=[ FirstEigenValue, SecondEigenValue, ThirdEigenValue];

NewMatrix3 := Matrix([[x1,x2,x3], [x4,x5,x6], [x7,x8,x9]]);

EigenvectorSol := simplify(solve({seq(seq((NewMatrix3 . NewInput3(1..-1,i))[j]=(v[i]* NewInput3(1..-1,i))[j], j=1..3), i=1..3)}, {seq(x||i, i=1..9)}));

EigenvectorT := Matrix([[rhs(EigenvectorSol[1]), rhs(EigenvectorSol[2]), rhs(EigenvectorSol[3])],[ rhs(EigenvectorSol[4]), rhs(EigenvectorSol[5]), rhs(EigenvectorSol[6])],[ rhs(EigenvectorSol[7]), rhs(EigenvectorSol[8]), rhs(EigenvectorSol[9])]]);

Old_Asso_eigenvector := Eigenvectors(MatrixMatrixMultiply(Transpose(InputMatrix3), InputMatrix3));

 

sys1:=MatrixMatrixMultiply(NewInput3-Matrix([[FirstEigenValue, 0, 0], [0, FirstEigenValue, 0], [0, 0, FirstEigenValue]]),Matrix([[x],[y],[z]]));

sys1a := NewInput3-Matrix([[FirstEigenValue, 0, 0], [0, FirstEigenValue, 0], [0, 0, FirstEigenValue]]);

 

fsolve({sys1[1][1]=0,sys1[2][1]=0,sys1[3][1]=0}, {x,y,z});

solve([sys1[1][1]=0,sys1[2][1]=0,sys1[3][1]=0], [x,y,z]);

solve([sys1[1]=0,sys1[2]=0,sys1[3]=0], [x,y,z]);

> solve([sys1[1] = 0, sys1[2] = 0, sys1[3] = 0], [x, y, z]);

Error, invalid input: solve expects its 1st argument, eqs, to be of type {`and`, `not`, `or`, algebraic, relation(algebraic), ({set, list})({`and`, `not`, `or`, algebraic, relation(algebraic)})}, but received [(Vector[row](1, {(1) = HFloat(2571.1332294000003)*x+HFloat(1901.9)*y+HFloat(953.125)*z})) = 0, (Vector[row](1, {(1) = HFloat(1901.9)*x+HFloat(1594.5707294000001)*y+HFloat(939.4)*z})) = 0, (Vector[row](1, {(1) = HFloat(953.125)*x+HFloat(939.4)*y+HFloat(645.9307294)*z})) = 0]

 

 

v1 := <sys1a[1,1] | sys1a[1,2] | sys1a[1,3]>;

v2 := <sys1a[2,1] | sys1a[2,2] | sys1a[2,3]>;

v3 := <sys1a[3,1] | sys1a[3,2] | sys1a[3,3]>;

 

v1 := <sys1a[1,1] | sys1a[2,1] | sys1a[3,1]>;

v2 := <sys1a[1,2] | sys1a[2,2] | sys1a[3,2]>;

v3 := <sys1a[1,3] | sys1a[2,3] | sys1a[3,3]>;

eigenvector1 := Basis([v1, v2, v2]);

eliminate({sys1[1][1]=0,sys1[2][1]=0,sys1[3][1]=0},{x,y,z});

eliminate({sys1[1][1]=0,sys1[1][2]=0,sys1[1][3]=0},{x,y,z});

 

sys1:=MatrixMatrixMultiply(NewInput3-Matrix([[SecondEigenValue, 0, 0], [0, SecondEigenValue, 0], [0, 0, SecondEigenValue]]),Matrix([[x],[y],[z]]));

solve([sys1[1][1]=0,sys1[2][1]=0,sys1[3][1]=0], [x,y,z]);

sys1:=MatrixMatrixMultiply(NewInput3-Matrix([[ThirdEigenValue, 0, 0], [0, ThirdEigenValue, 0], [0, 0, ThirdEigenValue]]),Matrix([[x],[y],[z]]));

solve([sys1[1][1]=0,sys1[2][1]=0,sys1[3][1]=0], [x,y,z]);

 

https://drive.google.com/file/d/0B2D69u2pweEvUDJIeGlOVjFvNWc/edit?usp=sharing
https://drive.google.com/file/d/0B2D69u2pweEvV1BiRXhULTNPcWM/edit?usp=sharing
https://drive.google.com/file/d/0B2D69u2pweEvdXNrRlNadldXS0U/edit?usp=sharing

i find that maple 15 values are the same as extreme optimization library however, the sign are different

is it maple 15 accuracy correct or extreme library correct?

https://drive.google.com/file/d/0B2D69u2pweEvT01pazBxOEk1bWc/edit?usp=sharing

i worry for my research whether based on correct accuracy.

 

it can run without error in maple 15, however, the eigenvector values are wrong in maple 15 different from eigenvector function's result

then i test it in maple 12, it got error when run with following input

Warning, solutions may have been lost
Error, invalid input: simplify uses a 1st argument, s, which is missing
> InputMatrix3;
                           [[30.15,29.95,29.95],[29.95,29.95,0],[29.95,0,0]]

NewInput3 := MatrixMatrixMultiply(Transpose(InputMatrix3), InputMatrix3);
FirstEigenValue := solve(Determinant(NewInput3-Matrix([[lambda1, 0, 0], [0, lambda1, 0], [0, 0, lambda1]])), lambda1)[1]; # find back eigenvalue from eigenvector
SecondEigenValue := solve(Determinant(NewInput3-Matrix([[lambda1, 0, 0], [0, lambda1, 0], [0, 0, lambda1]])), lambda1)[2]; # find back eigenvalue from eigenvector
ThirdEigenValue := solve(Determinant(NewInput3-Matrix([[lambda1, 0, 0], [0, lambda1, 0], [0, 0, lambda1]])), lambda1)[3]; # find back eigenvalue from eigenvector
v:=[ FirstEigenValue, SecondEigenValue, ThirdEigenValue];
NewMatrix3 := Matrix([[x1,x2,x3], [x4,x5,x6], [x7,x8,x9]]);
EigenvectorSol := simplify(solve({seq(seq((NewMatrix3 . NewInput3(1..-1,i))[j]=(v[i]* NewInput3(1..-1,i))[j], j=1..3), i=1..3)}, {seq(x||i, i=1..9)}));
EigenvectorT := Matrix([[rhs(EigenvectorSol[1]), rhs(EigenvectorSol[2]), rhs(EigenvectorSol[3])],[ rhs(EigenvectorSol[4]), rhs(EigenvectorSol[5]), rhs(EigenvectorSol[6])],[ rhs(EigenvectorSol[7]), rhs(EigenvectorSol[8]), rhs(EigenvectorSol[9])]]);
Old_Asso_eigenvector := Eigenvectors(MatrixMatrixMultiply(Transpose(InputMatrix3), InputMatrix3));

with(LinearAlgebra):
t:=1;
NewMatrix3 := Matrix([[test10, close3(t) , close3(t+1)],
[close3(t) , close3(t+1) ,0],
[close3(t+1) , 0,0]]);

Matrix(3, 3, {(1, 1) = test10, (1, 2) = 5.59, (1, 3) = 5.74, (2, 1) = 5.59, (2, 2) = 5.74, (2, 3) = 0, (3, 1) = 5.74, (3, 2) = 0, (3, 3) = 0})

NewEigenMatrix := Eigenvalues(NewMatrix3);
solve([MatrixMatrixMultiply(NewMatrix3,Matrix([[x],[y],[z]]))[1][1] = NewEigenMatrix[1]* Matrix([[x],[y],[z]])[1][1],
MatrixMatrixMultiply(NewMatrix3,Matrix([[x],[y],[z]]))[2][1] = NewEigenMatrix[1]* Matrix([[x],[y],[z]])[2][1],
MatrixMatrixMultiply(NewMatrix3,Matrix([[x],[y],[z]]))[3][1] = NewEigenMatrix[1]* Matrix([[x],[y],[z]])[3][1]]
, [x,y,z]);

expect to calculate a eigenvector in terms of variable test10

close3 are decimal value

> NewMatrix3 := Matrix([[test10, close3(t), close3(t+1), close3(t+2), close3(t+3), close3(t+4)], [close3(t), close3(t+1), close3(t+2), close3(t+3), close3(t+4), close3(t+5)], [close3(t+1), close3(t+2), close3(t+3), close3(t+4), close3(t+5), 0], [close3(t+2), close3(t+3), close3(t+4), close3(t+5), 0, 0], [close3(t+3), close3(t+4), close3(t+5), 0, 0, 0], [close3(t+4), close3(t+5), 0, 0, 0, 0], [close3(t+5), 0, 0, 0, 0, 0]]); New_Asso_eigenvector := Eigenvectors(MatrixMatrixMultiply(Transpose(NewMatrix3), NewMatrix3));


Error, (in LA_Main:-Eigenvectors) cannot determine if this expression is true or false: abs(149.8198+5.59*Re(test10))+27.38*abs(Im(test10))+abs(118.8174+5.74*Re(test10))+abs(90.3603+5.49*Re(test10))+abs(61.9327+5.19*Re(test10))+abs(31.0804+5.37*Re(test10)) < (1/10)*abs(149.8198+5.59*Re(test10))+2.738000000*abs(Im(test10))+(1/10)*abs(118.8174+5.74*Re(test10))+(1/10)*abs(90.3603+5.49*Re(test10))+(1/10)*abs(61.9327+5.19*Re(test10))+(1/10)*abs(31.0804+5.37*Re(test10))

 

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:

 


Hey,

I have currently encountered a problem that I am not sure if is a mathematical problem or a problem with Maple itself. I want to find the eigenvalues and eigenvectors of the following matrix:

[[2.460*10^9*sin(theta)^2+8.970*10^6*cos(theta)^2, 0, 2.449*10^9*sin(theta)*cos(theta)], [0,8.450*10^6*sin(theta)^2+8.970*10^6*cos(theta)^2,0],[2.449*10^9*sin(theta)*cos(theta),8.970*10^6*sin(theta)^2+2.530*10^9*cos(theta)^2]]

Maple is able to conjure...

mech_problem.mw

Hi

I solve a mechanical exercise but i had a problem.

I know M (mass) and S (stifness) matrices (6x6).

I want to solve the (λ2M+S)v=0  eigenvalue problem, where λ are the eigenvalues and

Hi, I have a matrix D which is a 33x33 matrix. I find the eigenvector using "Eigenvectors(D)". It gives me the eigenvalue together with the eigenvector. Then, i want to multiply the eignevector with another 33x33 matrix. How can I get the eigenvector without the eigenvalues attached to it? please advice.

Many thanks.

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