How to Handle Zero Eigenvalues in the Generalized Eigenvalue Problem?

In summary, the user is trying to find the λ and ai that solves a Generalized eigenvalue problem with given matrices A and B. Using Mathematica, the user found the lambdas and noticed that the last two eigenvalues are zero, indicating that two equations are redundant. However, this is not an issue and has a physical interpretation. The user is looking for a way to reduce the system before calculating the eigenvalues to only have nontrivial eigenvalues.
  • #1
member 399911
Hi all,

I need to find the λ and the ai that solves the Generalized eigenvalue problem

[A]{a}=-λ2 {a}

with

[A]=
Code:
-1289.57,1204.12,92.5424,-7.09489,-25037.4,32022.5,-10004.3,3019.17
1157.46,-1077.94,-0.580522,-78.9482,32022.5,-57353.5,36280.6,-10949.6
166.577,-103.776,1494.41,-1557.21,-10004.3,36280.6,-63053.2,36776.9
-34.4753,-22.407,-1586.37,1643.25,3019.17,-10949.6,36776.9,-28846.5
-22840.1,29254.3,-9328.5,2914.31,-1289.57,1157.46,166.577,-34.4753
29254.3,-51067,31724.8,-9912.15,1204.12,-1077.94,-103.776,-22.407
-9328.5,31724.8,-54591.9,32195.6,92.5424,-0.580522,1494.41,-1586.37
2914.31,-9912.15,32195.6,-25197.7,-7.09489,-78.9482,-1557.21,1643.25

and

=

Code:
0,0,0,0,1875.81,0,0,0
0,5019.07,0,0,0,22535.3,0,0
0,0,-5019.07,0,0,0,22535.3,0
0,0,0,0,0,0,0,937.905
835.2,0,0,0,0,0,0,0
0,5003.02,0,0,0,5019.07,0,0
0,0,5003.02,0,0,0,-5019.07,0
0,0,0,417.6,0,0,0,0

Using mathematica I get for lambdas {75.1098, 35.2687, 34.3082, 15.2013, 4.3281, 1.35478,
5.38827*10^-154, -2.06904*10^-154}

The last two eigenvalues are zero. That made me think that two equations are redundant. I confirmed that with mathematica. Rank of A is 6.

How can I reduce the system? I tried deleting two rows and columns, but for all the combinations the matrix B is singular.
Notice that the first and last column/row of the matrix B have only one value different from zero. Is anything I can do to exploit that?

Best
 
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  • #2
Why do you think having two zero eigenvalues is an issue? If A has rank 6 and B has rank 4, that is what you would expect to get.

For example if A was the mass matrix and B the stiffness matrix of a structure, the two zero eigenvalues just mean the system can move freely in two (generalized) directions. There is nothing "unphysical" or unreasonable about that.

You don't say what your matrices represent, but if they come from a physics problem the zero eigenvalues probably have a physical interpretation.
 
  • #3
AlephZero said:
Why do you think having two zero eigenvalues is an issue? If A has rank 6 and B has rank 4, that is what you would expect to get.

For example if A was the mass matrix and B the stiffness matrix of a structure, the two zero eigenvalues just mean the system can move freely in two (generalized) directions. There is nothing "unphysical" or unreasonable about that.

You don't say what your matrices represent, but if they come from a physics problem the zero eigenvalues probably have a physical interpretation.

Thanks AlephZero. The zero eigenvalues do have a physical interpretation and it is related to what you said.

Those special cases are treated separately and I do not need the associated eigenvectors.

What I do up to now is solve the 8x8 system and then remove the zero eigenvalues. I want to know if there is a way reduce the system before calculating the eigenvalues so as to have the nontrivial eigenvalues.

Best
 

FAQ: How to Handle Zero Eigenvalues in the Generalized Eigenvalue Problem?

1. What is a generalized eigenvalue problem?

A generalized eigenvalue problem is a mathematical problem that involves finding a set of scalar values, called eigenvalues, and corresponding non-zero vectors, called eigenvectors, that satisfy the equation Ax = λBx, where A and B are square matrices of the same size. This problem has many applications in various fields, such as physics, engineering, and computer science.

2. How is a generalized eigenvalue problem different from a regular eigenvalue problem?

A regular eigenvalue problem involves finding the eigenvalues and eigenvectors of a single square matrix, while a generalized eigenvalue problem involves finding the eigenvalues and eigenvectors of two square matrices. In a regular eigenvalue problem, the matrix B is the identity matrix, so the equation becomes Ax = λx. In a generalized eigenvalue problem, both A and B matrices are involved, making it a more complex problem to solve.

3. What are the applications of a generalized eigenvalue problem?

A generalized eigenvalue problem has many applications in various fields, such as signal processing, control systems, quantum mechanics, and structural engineering. It is used to solve problems involving the behavior of vibrating systems, stability analysis of control systems, and the study of quantum mechanical systems.

4. How is a generalized eigenvalue problem solved?

There are various methods for solving a generalized eigenvalue problem, such as the QR algorithm, the Lanczos algorithm, and the Jacobi-Davidson method. These methods involve iterative calculations and use properties of the matrices A and B to find the eigenvalues and eigenvectors. The choice of method depends on the size and structure of the matrices and the desired accuracy of the solution.

5. What are the advantages of using a generalized eigenvalue problem in scientific research?

A generalized eigenvalue problem allows for a deeper understanding of the behavior of complex systems and can provide insights into the underlying structures and relationships within the data. It also has a wide range of applications in various fields, making it a valuable tool for scientists and researchers. Additionally, the solutions obtained from a generalized eigenvalue problem can be used to make predictions and inform decision-making in real-world situations.

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