Items tagged with eigenvectors

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I solve a mechanical exercise but i had a problem.

I know M (mass) and K (stifness) matrices (4x4).

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

How can i solve this problem.  I tried with the Eigenvectors() command but it didn't give the right solution.

The Eigenvalues are okay, but the eigenvectors not

K := Matrix([[4*10^7,-1.50*10^7,2*10^7,0],[-1.50*10^7,1.50*10^7,0,1.50*10^7],[2*10^7,0,8*10^7,2*10^7],[0,1.50*10^7,2*10^7,4*10^7]]);

M:=Matrix([[121.90,99.048,-91.429,0],[99.048,594.29,0,-99.048],[-91.429,0,243.81,-91.429],[0,-99.048,-91.429,121.90]]);

w1,w2:=Eigenvectors(K,M);

Acoording with the book the right eigenvectors(shape mode) are:

[0.013 991,  0.034 233,  0.073 683,  0.090 573]
[0.035 637, 0, -0.032 213, 0]
[0 ,-0.034 233, 0, 0.090 573]
[-0.013 991, 0.034 233, -0.073 683, 0.090 573]

Thank you
 

hi.i cant write english very well. excuse me

i solve the Eigenvectors of two matrix. first i want to delete complex numbers in solve. and then sort the little vector in first matrix of vector. as this picture

1.mw

 

code

k__1 := (12*2)*10^6/3^3; k__2 := k__1; k__3 := (3*1.5)*10^6/3^3; k__4 := (12*1.5)*10^6/3^3; k__5 := 12*10^6/3^3; m__1 := 8000; m__2 := 7000; m__3 := 6000; m__4 := 5000; m__5 := 10000; K := Matrix(5, 5, [[k__1+k__2, -k__2, 0, 0, 0], [-k__2, k__2+k__3, -k__3, 0, 0], [0, -k__3, k__3+k__4, -k__4, 0], [0, 0, -k__4, k__4+k__5, -k__5], [0, 0, 0, -k__5, k__5]]); M := Matrix(5, 5, [[m__1, 0, 0, 0, 0], [0, m__2, 0, 0, 0], [0, 0, m__3, 0, 0], [0, 0, 0, m__4, 0], [0, 0, 0, 0, m__5]])

In Maple18.02:

Hso := Matrix(8, {(1, 4) = -x, (1, 6) = I*x, (2, 3) = x, (2, 5) = I*x, (3, 2) = x, (3, 5) = -I*z, (3, 8) = y, (4, 1) = -x, (4, 6) = I*z, (4, 7) = -y, (5, 2) = -I*x, (5, 3) = I*z, (5, 8) = -I*y, (6, 1) = -I*x, (6, 4) = -I*z, (6, 7) = -I*y, (7, 4) = -y, (7, 6) = I*y, (8, 3) = y, (8, 5) = I*y})

av, AV := LinearAlgebra[Eigenvectors](Hso)

Error, (in Polynomial:-Quadratic) type `truefalseFAIL` does not exist


This does not happen in Maple17.

hi.i have a problem with gain Eigenvectors

please help me

thanks

lorenz_1.mw

restart; with(LinearAlgebra[Modular]); A := Matrix(3, 3, [[-a, a, 0], [0, 0, -sqrt(l.b.c.(h+k))/(h+k)], [(2.*h.sqrt(l.b.c.(h+k)))/(l.(h+k)), (2.*k.sqrt(l.b.c.(h+k)))/(l.(h+k)), -c]])

A := Matrix(3, 3, {(1, 1) = -a, (1, 2) = a, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = -(`.`(l, b, c, h+k))^(1/2)/(h+k), (3, 1) = 2.*(h.((`.`(l, b, c, h+k))^(1/2)))/(l.(h+k)), (3, 2) = 2.*(k.((`.`(l, b, c, h+k))^(1/2)))/(l.(h+k)), (3, 3) = -c})

(1)

Sol := LinearAlgebra[Eigenvectors](A)

Error, (in evala/preproc3) floats not handled yet

 

``


Download lorenz_1.mw

Hello! 

In an assigment I have been asked to use the Eigenvectors command to find the eigenvalues and eigenvectors of a particular matrix. 

As highlighted in the following image, my questions are:

1. What is the meaning of the suffix "+0.I"? Does it mean that there are further decimal digits which are not displayed?

2. How do the first and third eigenvalues, which are equal, result in different eigenvectors? As per my understanding, equal eigenvalues should have equal corresponding eigenvectors. Please help.

nullspace or reducedform or Eigenvectors still can not find eigenvector in terms of  mmm , how to find this?

 

mmm is a variable

 in eigenvector using nullspace and eigenvector using maple function  Eigenvectors ? 

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?

in LinearAlgebra Eigenvectors calculation.

Maple 2015 Error

 

 

So the above output startled me.  I have used the Maple Linear Algebra Eigenvalues, Eigenvectors commands many times with no problem.  Can any one explain to me what is going on.  The program correctly calculates the eigenvalues for the matrix which are all distinct for a real symmetric matrix, and thus should have three distinct non-zero eigenvectors, yet the eigenvectore command returns only zeros for the eigenvectors.  I calculated an eigenvector by hand corresponding to the eigenvalue of 1 and obtained (1, -sqrt(2)/sqrt(3), -1/sqrt(3).

 

So this is either a serious bug or I am going completely insane. 

I am getting different results when determining the eigenvectors of a simple beam vibration problem in 2 ways :

See attachment :
1) When generating a matrix with GenerateMatrix([eqs..],[A,B...]) , or with
2) reformulating the same matrix by hand with S:=Matrix(4,4,[[ ]]) I get different eigenvectors for actually the same matrices ?

Tried already commands like 'convert' , 'Normalize', setting precision, to no avail.
Is this due to rounding ?

Any suggestions ?

Thanks ! 

Eigenvectors_2.mw

 

Hi there

I have have a 18*18 matrix which almost each of its element are in symbolic form. Now I need to have all of its eigenvectors. Unfortunately when I use the "Eigenvalues()" function in maple i got nothing. In fact I got the error which comes below.

Error, (in content/polynom) general case of floats not handled

I need to know if there's a solution to eliminate the error? If not, what can I do to determine the eigenvectors and eigenvalues in symbolic form?

I'll be appreciated your help

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));

superposition said that a vector is a linear combination of other vectors

but even if i calculated the coefficient, i do not know which vector is which other vectors's linear combination

how to prove?

InputMatrix3 := Matrix([[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) , close3(t+6)],
[close3(t+2) , close3(t+3) , close3(t+4) , close3(t+5) , close3(t+6) , 0],
[close3(t+3) , close3(t+4) , close3(t+5) , close3(t+6) , 0 , 0],
[close3(t+4) , close3(t+5) , close3(t+6) , 0 , 0 , 0],
[close3(t+5) , close3(t+6) , 0 , 0 , 0, 0],
[close3(t+6) , 0 , 0 , 0, 0, 0]]):
EigenValue1 := Eigenvalues(MatrixMatrixMultiply(Transpose(InputMatrix3), InputMatrix3)):
Asso_eigenvector := Eigenvectors(MatrixMatrixMultiply(Transpose(InputMatrix3), InputMatrix3)):
AEigenVector[tt+1] := Asso_eigenvector;

Matrix(6, 6, {(1, 1) = .514973850028629+0.*I, (1, 2) = .510603608194333+0.*I, (1, 3) = .469094659512372+0.*I, (1, 4) = .389872713818831+0.*I, (1, 5) = .279479324327359+0.*I, (1, 6) = -.154682461176604+0.*I, (2, 1) = .493994413154560+0.*I, (2, 2) = .306651336822139+0.*I, (2, 3) = -0.583656699197969e-1+0.*I, (2, 4) = -.417550308930506+0.*I, (2, 5) = -.566122865008542+0.*I, (2, 6) = .404579494288380+0.*I, (3, 1) = .449581541124671+0.*I, (3, 2) = -0.266751368453398e-1+0.*I, (3, 3) = -.529663398913996+0.*I, (3, 4) = -.359719616523673+0.*I, (3, 5) = .313717798014566+0.*I, (3, 6) = -.537405340038665+0.*I, (4, 1) = .386952162293470+0.*I, (4, 2) = -.351332186748244+0.*I, (4, 3) = -.390816901794187+0.*I, (4, 4) = .470032416161955+0.*I, (4, 5) = .231969182174424+0.*I, (4, 6) = .547134073332474+0.*I, (5, 1) = .306149178348317+0.*I, (5, 2) = -.530611390076568+0.*I, (5, 3) = .192717713961280+0.*I, (5, 4) = .291213691618787+0.*I, (5, 5) = -.562991429686901+0.*I, (5, 6) = -.431067688369314+0.*I, (6, 1) = .212576094920847+0.*I, (6, 2) = -.489443150196337+0.*I, (6, 3) = .553283259136031+0.*I, (6, 4) = -.488381938231088+0.*I, (6, 5) = .363604594054259+0.*I, (6, 6) = .195982711855368+0.*I})

Matrix(6, 6, {(1, 1) = .515428842592397+0.*I, (1, 2) = .515531996615269+0.*I, (1, 3) = .468108280940919+0.*I, (1, 4) = -.392394120975052+0.*I, (1, 5) = -.280467124908196+0.*I, (1, 6) = -.129613084502380+0.*I, (2, 1) = .494563493180197+0.*I, (2, 2) = .301273494494509+0.*I, (2, 3) = -0.622136916501293e-1+0.*I, (2, 4) = .438383262732459+0.*I, (2, 5) = .571041594120088+0.*I, (2, 6) = .377494770878435+0.*I, (3, 1) = .450886315308369+0.*I, (3, 2) = -0.323387895921418e-1+0.*I, (3, 3) = -.527636820417566+0.*I, (3, 4) = .332744872607714+0.*I, (3, 5) = -.322934536375586+0.*I, (3, 6) = -.549772001891837+0.*I, (4, 1) = .385916641681991+0.*I, (4, 2) = -.352066020655722+0.*I, (4, 3) = -.389655495441319+0.*I, (4, 4) = -.450049711766943+0.*I, (4, 5) = -.221529986447276+0.*I, (4, 6) = .568916672007495+0.*I, (5, 1) = .305485655770791+0.*I, (5, 2) = -.528766119966973+0.*I, (5, 3) = .201065789602278+0.*I, (5, 4) = -.310329356773806+0.*I, (5, 5) = .555973984740943+0.*I, (5, 6) = -.425730045170186+0.*I, (6, 1) = .210210489500614+0.*I, (6, 2) = -.488744465076970+0.*I, (6, 3) = .553484076328700+0.*I, (6, 4) = .494245653290329+0.*I, (6, 5) = -.364390406353340+0.*I, (6, 6) = .183130120876843+0.*I})
mm1 := 1;
solve(
[AEigenVector[mm1][2][1][6] = m1*AEigenVector[mm1][2][1][1]+m2*AEigenVector[mm1][2][1][2]+m3*AEigenVector[mm1][2][1][3]+m4*AEigenVector[mm1][2][1][4]+m5*AEigenVector[mm1][2][1][5],
AEigenVector[mm1][2][2][6] = m1*AEigenVector[mm1][2][2][1]+m2*AEigenVector[mm1][2][2][2]+m3*AEigenVector[mm1][2][2][3]+m4*AEigenVector[mm1][2][2][4]+m5*AEigenVector[mm1][2][2][5],
AEigenVector[mm1][2][3][6] = m1*AEigenVector[mm1][2][3][1]+m2*AEigenVector[mm1][2][3][2]+m3*AEigenVector[mm1][2][3][3]+m4*AEigenVector[mm1][2][3][4]+m5*AEigenVector[mm1][2][3][5],
AEigenVector[mm1][2][4][6] = m1*AEigenVector[mm1][2][4][1]+m2*AEigenVector[mm1][2][4][2]+m3*AEigenVector[mm1][2][4][3]+m4*AEigenVector[mm1][2][4][4]+m5*AEigenVector[mm1][2][4][5],
m1^2 + m2^2 + m3^2 + m4^2 + m5^2 = 1], [m1, m2, m3, m4, m5]);

[m1 = .4027576723+.5022235499*I, m2 = -.5922841426-1.043213223*I, m3 = -.1130969773+.9150300317*I, m4 = .9867039883-.5082455178*I, m5 = -1.400123192+.1536850673*I], [m1 = .4027576723-.5022235499*I, m2 = -.5922841426+1.043213223*I, m3 = -.1130969773-.9150300317*I, m4 = .9867039883+.5082455178*I, m5 = -1.400123192-.1536850673*I]

mm1 := 2;
solve(
[AEigenVector[mm1][2][1][6] = m1*AEigenVector[mm1][2][1][1]+m2*AEigenVector[mm1][2][1][2]+m3*AEigenVector[mm1][2][1][3]+m4*AEigenVector[mm1][2][1][4]+m5*AEigenVector[mm1][2][1][5],
AEigenVector[mm1][2][2][6] = m1*AEigenVector[mm1][2][2][1]+m2*AEigenVector[mm1][2][2][2]+m3*AEigenVector[mm1][2][2][3]+m4*AEigenVector[mm1][2][2][4]+m5*AEigenVector[mm1][2][2][5],
AEigenVector[mm1][2][3][6] = m1*AEigenVector[mm1][2][3][1]+m2*AEigenVector[mm1][2][3][2]+m3*AEigenVector[mm1][2][3][3]+m4*AEigenVector[mm1][2][3][4]+m5*AEigenVector[mm1][2][3][5],
AEigenVector[mm1][2][4][6] = m1*AEigenVector[mm1][2][4][1]+m2*AEigenVector[mm1][2][4][2]+m3*AEigenVector[mm1][2][4][3]+m4*AEigenVector[mm1][2][4][4]+m5*AEigenVector[mm1][2][4][5],
m1^2 + m2^2 + m3^2 + m4^2 + m5^2 = 1], [m1, m2, m3, m4, m5]);

[m1 = .4262845394-.5114193433*I, m2 = -.6313720018+1.072185334*I, m3 = -0.7337582213e-1-.9580760394*I, m4 = -1.036525681-.5400714113*I, m5 = 1.412710014+.1874839516*I], [m1 = .4262845394+.5114193433*I, m2 = -.6313720018-1.072185334*I, m3 = -0.7337582213e-1+.9580760394*I, m4 = -1.036525681+.5400714113*I, m5 = 1.412710014-.1874839516*I]

Heey

 

I have a problem when I call the Eigenvectors i get:

>Eigenvectors(Transpose(K).K);

i get the answer i want but every time i execute the worksheet my eigenvalues swich places aswell as the eigenvector.

Someone that can help me how i can sort or something like that to freeze the eigenvector output? so i can use some of the output to caluculate furhtere on and when i execute the worksheet i know that me eigenvectors and eigenvalues...

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