Unable to evaluate Eigenvalues and Eigenvectors for a matrix (2)












2












$begingroup$


I have posted a similar question last year pertaining to this issue. Here's a link to my post together with the solution given: Unable to evaluate Eigenvalues and Eigenvectors for a matrix



I have tried the methods in my previous posts but to no avail. Here's the problem: I have the following 3x3 matrix



m = {{-γ/2, -I*g1, -I*Exp[-I*α]*g3}, {-I*g1, -(κ1)/2, -I*g2}, {-I*Exp[I*α]*g3, -I*g2, -(κ2)/2]}}


where I represents the complex identity Sqrt[-1]. I wish to find the eigenvectors for the matrix for two different alpha values. For α = π/2, simply doing (after manually replacing α with π/2)



Eigenvectors[m, Cubics->True]


Returns the appropriate (albeit long) eigenvectors. Now however, if I change my α to α = π and run 



 Eigenvectors[m, Cubics->True]


I am returned with



...Eigenvectors: Unable to find all eigenvectors


Which is the similar issue encountered in the link that I provided above a while ago. I proceed to perform the same fix detailed in that question. Namely



Simplify[Eigenvectors[mchiral /. Complex[0, -1] -> mi, Cubics -> True] /. mi -> -I];


and I am still returned with the same error. Namely



...Eigenvectors: Unable to find all eigenvectors


What is the problem here?






matrix eigenvalues







share|improve this question











$endgroup$












  • $begingroup$
    There is no problem: imgur.com/a/ALdYCou Except that you have some brackets misplaced in the definition of m that I fixed – but check if the form is the desired one.
    $endgroup$
    – corey979
    4 hours ago












  • $begingroup$
    @corey979 Interesting. I am on version 11.3 for macOS and I do get the error message (even after fixing the brackets).
    $endgroup$
    – Henrik Schumacher
    4 hours ago










  • $begingroup$
    How many times have I advocated for providing the $Version one is using... I'm on 10.4 and there is no problem, as showed. Indeed, there is an error in 11.3. No idea what version the OP is using. Unless he clarifies there is virtually no problem.
    $endgroup$
    – corey979
    3 hours ago










  • $begingroup$
    @corey979 Apologies but I'm using version 11.3. The error still persists even after the bracket fix. I don't know what the problem is
    $endgroup$
    – kowalski
    3 hours ago
















2












$begingroup$


I have posted a similar question last year pertaining to this issue. Here's a link to my post together with the solution given: Unable to evaluate Eigenvalues and Eigenvectors for a matrix



I have tried the methods in my previous posts but to no avail. Here's the problem: I have the following 3x3 matrix



m = {{-γ/2, -I*g1, -I*Exp[-I*α]*g3}, {-I*g1, -(κ1)/2, -I*g2}, {-I*Exp[I*α]*g3, -I*g2, -(κ2)/2]}}


where I represents the complex identity Sqrt[-1]. I wish to find the eigenvectors for the matrix for two different alpha values. For α = π/2, simply doing (after manually replacing α with π/2)



Eigenvectors[m, Cubics->True]


Returns the appropriate (albeit long) eigenvectors. Now however, if I change my α to α = π and run 



 Eigenvectors[m, Cubics->True]


I am returned with



...Eigenvectors: Unable to find all eigenvectors


Which is the similar issue encountered in the link that I provided above a while ago. I proceed to perform the same fix detailed in that question. Namely



Simplify[Eigenvectors[mchiral /. Complex[0, -1] -> mi, Cubics -> True] /. mi -> -I];


and I am still returned with the same error. Namely



...Eigenvectors: Unable to find all eigenvectors


What is the problem here?






matrix eigenvalues







share|improve this question











$endgroup$












  • $begingroup$
    There is no problem: imgur.com/a/ALdYCou Except that you have some brackets misplaced in the definition of m that I fixed – but check if the form is the desired one.
    $endgroup$
    – corey979
    4 hours ago












  • $begingroup$
    @corey979 Interesting. I am on version 11.3 for macOS and I do get the error message (even after fixing the brackets).
    $endgroup$
    – Henrik Schumacher
    4 hours ago










  • $begingroup$
    How many times have I advocated for providing the $Version one is using... I'm on 10.4 and there is no problem, as showed. Indeed, there is an error in 11.3. No idea what version the OP is using. Unless he clarifies there is virtually no problem.
    $endgroup$
    – corey979
    3 hours ago










  • $begingroup$
    @corey979 Apologies but I'm using version 11.3. The error still persists even after the bracket fix. I don't know what the problem is
    $endgroup$
    – kowalski
    3 hours ago














2












2








2





$begingroup$


I have posted a similar question last year pertaining to this issue. Here's a link to my post together with the solution given: Unable to evaluate Eigenvalues and Eigenvectors for a matrix



I have tried the methods in my previous posts but to no avail. Here's the problem: I have the following 3x3 matrix



m = {{-γ/2, -I*g1, -I*Exp[-I*α]*g3}, {-I*g1, -(κ1)/2, -I*g2}, {-I*Exp[I*α]*g3, -I*g2, -(κ2)/2]}}


where I represents the complex identity Sqrt[-1]. I wish to find the eigenvectors for the matrix for two different alpha values. For α = π/2, simply doing (after manually replacing α with π/2)



Eigenvectors[m, Cubics->True]


Returns the appropriate (albeit long) eigenvectors. Now however, if I change my α to α = π and run 



 Eigenvectors[m, Cubics->True]


I am returned with



...Eigenvectors: Unable to find all eigenvectors


Which is the similar issue encountered in the link that I provided above a while ago. I proceed to perform the same fix detailed in that question. Namely



Simplify[Eigenvectors[mchiral /. Complex[0, -1] -> mi, Cubics -> True] /. mi -> -I];


and I am still returned with the same error. Namely



...Eigenvectors: Unable to find all eigenvectors


What is the problem here?






matrix eigenvalues







share|improve this question











$endgroup$




I have posted a similar question last year pertaining to this issue. Here's a link to my post together with the solution given: Unable to evaluate Eigenvalues and Eigenvectors for a matrix



I have tried the methods in my previous posts but to no avail. Here's the problem: I have the following 3x3 matrix



m = {{-γ/2, -I*g1, -I*Exp[-I*α]*g3}, {-I*g1, -(κ1)/2, -I*g2}, {-I*Exp[I*α]*g3, -I*g2, -(κ2)/2]}}


where I represents the complex identity Sqrt[-1]. I wish to find the eigenvectors for the matrix for two different alpha values. For α = π/2, simply doing (after manually replacing α with π/2)



Eigenvectors[m, Cubics->True]


Returns the appropriate (albeit long) eigenvectors. Now however, if I change my α to α = π and run 



 Eigenvectors[m, Cubics->True]


I am returned with



...Eigenvectors: Unable to find all eigenvectors


Which is the similar issue encountered in the link that I provided above a while ago. I proceed to perform the same fix detailed in that question. Namely



Simplify[Eigenvectors[mchiral /. Complex[0, -1] -> mi, Cubics -> True] /. mi -> -I];


and I am still returned with the same error. Namely



...Eigenvectors: Unable to find all eigenvectors


What is the problem here?






matrix eigenvalues




matrix eigenvalues






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited 4 hours ago









corey979

20.9k64282




20.9k64282










asked 4 hours ago









kowalskikowalski

1559




1559












  • $begingroup$
    There is no problem: imgur.com/a/ALdYCou Except that you have some brackets misplaced in the definition of m that I fixed – but check if the form is the desired one.
    $endgroup$
    – corey979
    4 hours ago












  • $begingroup$
    @corey979 Interesting. I am on version 11.3 for macOS and I do get the error message (even after fixing the brackets).
    $endgroup$
    – Henrik Schumacher
    4 hours ago










  • $begingroup$
    How many times have I advocated for providing the $Version one is using... I'm on 10.4 and there is no problem, as showed. Indeed, there is an error in 11.3. No idea what version the OP is using. Unless he clarifies there is virtually no problem.
    $endgroup$
    – corey979
    3 hours ago










  • $begingroup$
    @corey979 Apologies but I'm using version 11.3. The error still persists even after the bracket fix. I don't know what the problem is
    $endgroup$
    – kowalski
    3 hours ago


















  • $begingroup$
    There is no problem: imgur.com/a/ALdYCou Except that you have some brackets misplaced in the definition of m that I fixed – but check if the form is the desired one.
    $endgroup$
    – corey979
    4 hours ago












  • $begingroup$
    @corey979 Interesting. I am on version 11.3 for macOS and I do get the error message (even after fixing the brackets).
    $endgroup$
    – Henrik Schumacher
    4 hours ago










  • $begingroup$
    How many times have I advocated for providing the $Version one is using... I'm on 10.4 and there is no problem, as showed. Indeed, there is an error in 11.3. No idea what version the OP is using. Unless he clarifies there is virtually no problem.
    $endgroup$
    – corey979
    3 hours ago










  • $begingroup$
    @corey979 Apologies but I'm using version 11.3. The error still persists even after the bracket fix. I don't know what the problem is
    $endgroup$
    – kowalski
    3 hours ago
















$begingroup$
There is no problem: imgur.com/a/ALdYCou Except that you have some brackets misplaced in the definition of m that I fixed – but check if the form is the desired one.
$endgroup$
– corey979
4 hours ago






$begingroup$
There is no problem: imgur.com/a/ALdYCou Except that you have some brackets misplaced in the definition of m that I fixed – but check if the form is the desired one.
$endgroup$
– corey979
4 hours ago














$begingroup$
@corey979 Interesting. I am on version 11.3 for macOS and I do get the error message (even after fixing the brackets).
$endgroup$
– Henrik Schumacher
4 hours ago




$begingroup$
@corey979 Interesting. I am on version 11.3 for macOS and I do get the error message (even after fixing the brackets).
$endgroup$
– Henrik Schumacher
4 hours ago












$begingroup$
How many times have I advocated for providing the $Version one is using... I'm on 10.4 and there is no problem, as showed. Indeed, there is an error in 11.3. No idea what version the OP is using. Unless he clarifies there is virtually no problem.
$endgroup$
– corey979
3 hours ago




$begingroup$
How many times have I advocated for providing the $Version one is using... I'm on 10.4 and there is no problem, as showed. Indeed, there is an error in 11.3. No idea what version the OP is using. Unless he clarifies there is virtually no problem.
$endgroup$
– corey979
3 hours ago












$begingroup$
@corey979 Apologies but I'm using version 11.3. The error still persists even after the bracket fix. I don't know what the problem is
$endgroup$
– kowalski
3 hours ago




$begingroup$
@corey979 Apologies but I'm using version 11.3. The error still persists even after the bracket fix. I don't know what the problem is
$endgroup$
– kowalski
3 hours ago










1 Answer
1






active

oldest

votes


















2












$begingroup$

I have o clue why this did not work. However, this old-fashioned method seems to work



m = {

     {-γ/2, -I g1, -I Exp[-I α] g3}, 

     {-I g1, -(κ1)/2, -I g2}, 

     {-I Exp[I α] g3, -I g2, -(κ2)/2}

     };



a = m /. α -> π;

λ = Eigenvalues[a] // ToRadicals;

U = Flatten[NullSpace[a - # IdentityMatrix[3]] & /@ λ, 1];



Simplify[a.Transpose[U] - Transpose[U].DiagonalMatrix[λ]]



{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}







share|improve this answer











$endgroup$













  • $begingroup$
    The eigenvalues seems to be fine. It's long but it prints. The eigenvectors on the other hand, are still unobtainable. I'm starting to think if this is alpha dependent since it works for pi/2 but not pi. But that's just silly and shouldn't happen
    $endgroup$
    – kowalski
    3 hours ago










  • $begingroup$
    @kowalski U produced by the code above contains the eigenvectors.
    $endgroup$
    – Henrik Schumacher
    3 hours ago











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1 Answer
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1 Answer
1






active

oldest

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active

oldest

votes






active

oldest

votes









2












$begingroup$

I have o clue why this did not work. However, this old-fashioned method seems to work



m = {

     {-γ/2, -I g1, -I Exp[-I α] g3}, 

     {-I g1, -(κ1)/2, -I g2}, 

     {-I Exp[I α] g3, -I g2, -(κ2)/2}

     };



a = m /. α -> π;

λ = Eigenvalues[a] // ToRadicals;

U = Flatten[NullSpace[a - # IdentityMatrix[3]] & /@ λ, 1];



Simplify[a.Transpose[U] - Transpose[U].DiagonalMatrix[λ]]



{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}







share|improve this answer











$endgroup$













  • $begingroup$
    The eigenvalues seems to be fine. It's long but it prints. The eigenvectors on the other hand, are still unobtainable. I'm starting to think if this is alpha dependent since it works for pi/2 but not pi. But that's just silly and shouldn't happen
    $endgroup$
    – kowalski
    3 hours ago










  • $begingroup$
    @kowalski U produced by the code above contains the eigenvectors.
    $endgroup$
    – Henrik Schumacher
    3 hours ago
















2












$begingroup$

I have o clue why this did not work. However, this old-fashioned method seems to work



m = {

     {-γ/2, -I g1, -I Exp[-I α] g3}, 

     {-I g1, -(κ1)/2, -I g2}, 

     {-I Exp[I α] g3, -I g2, -(κ2)/2}

     };



a = m /. α -> π;

λ = Eigenvalues[a] // ToRadicals;

U = Flatten[NullSpace[a - # IdentityMatrix[3]] & /@ λ, 1];



Simplify[a.Transpose[U] - Transpose[U].DiagonalMatrix[λ]]



{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}







share|improve this answer











$endgroup$













  • $begingroup$
    The eigenvalues seems to be fine. It's long but it prints. The eigenvectors on the other hand, are still unobtainable. I'm starting to think if this is alpha dependent since it works for pi/2 but not pi. But that's just silly and shouldn't happen
    $endgroup$
    – kowalski
    3 hours ago










  • $begingroup$
    @kowalski U produced by the code above contains the eigenvectors.
    $endgroup$
    – Henrik Schumacher
    3 hours ago














2












2








2





$begingroup$

I have o clue why this did not work. However, this old-fashioned method seems to work



m = {

     {-γ/2, -I g1, -I Exp[-I α] g3}, 

     {-I g1, -(κ1)/2, -I g2}, 

     {-I Exp[I α] g3, -I g2, -(κ2)/2}

     };



a = m /. α -> π;

λ = Eigenvalues[a] // ToRadicals;

U = Flatten[NullSpace[a - # IdentityMatrix[3]] & /@ λ, 1];



Simplify[a.Transpose[U] - Transpose[U].DiagonalMatrix[λ]]



{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}







share|improve this answer











$endgroup$



I have o clue why this did not work. However, this old-fashioned method seems to work



m = {

     {-γ/2, -I g1, -I Exp[-I α] g3}, 

     {-I g1, -(κ1)/2, -I g2}, 

     {-I Exp[I α] g3, -I g2, -(κ2)/2}

     };



a = m /. α -> π;

λ = Eigenvalues[a] // ToRadicals;

U = Flatten[NullSpace[a - # IdentityMatrix[3]] & /@ λ, 1];



Simplify[a.Transpose[U] - Transpose[U].DiagonalMatrix[λ]]



{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}








share|improve this answer














share|improve this answer



share|improve this answer








edited 3 hours ago

























answered 4 hours ago









Henrik SchumacherHenrik Schumacher

56.7k577157




56.7k577157












  • $begingroup$
    The eigenvalues seems to be fine. It's long but it prints. The eigenvectors on the other hand, are still unobtainable. I'm starting to think if this is alpha dependent since it works for pi/2 but not pi. But that's just silly and shouldn't happen
    $endgroup$
    – kowalski
    3 hours ago










  • $begingroup$
    @kowalski U produced by the code above contains the eigenvectors.
    $endgroup$
    – Henrik Schumacher
    3 hours ago


















  • $begingroup$
    The eigenvalues seems to be fine. It's long but it prints. The eigenvectors on the other hand, are still unobtainable. I'm starting to think if this is alpha dependent since it works for pi/2 but not pi. But that's just silly and shouldn't happen
    $endgroup$
    – kowalski
    3 hours ago










  • $begingroup$
    @kowalski U produced by the code above contains the eigenvectors.
    $endgroup$
    – Henrik Schumacher
    3 hours ago
















$begingroup$
The eigenvalues seems to be fine. It's long but it prints. The eigenvectors on the other hand, are still unobtainable. I'm starting to think if this is alpha dependent since it works for pi/2 but not pi. But that's just silly and shouldn't happen
$endgroup$
– kowalski
3 hours ago




$begingroup$
The eigenvalues seems to be fine. It's long but it prints. The eigenvectors on the other hand, are still unobtainable. I'm starting to think if this is alpha dependent since it works for pi/2 but not pi. But that's just silly and shouldn't happen
$endgroup$
– kowalski
3 hours ago












$begingroup$
@kowalski U produced by the code above contains the eigenvectors.
$endgroup$
– Henrik Schumacher
3 hours ago




$begingroup$
@kowalski U produced by the code above contains the eigenvectors.
$endgroup$
– Henrik Schumacher
3 hours ago


















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