Finding Eigenvectors and Values of Non-Hermitian Matrices with Mathematica

In summary, the conversation revolves around finding the eigenvectors and values of a non-Hermitian matrix in Mathematica. The person has been attempting to use the householder transformation method but has not seen any changes after 10 applications. They are looking for guidance and also questioning if Mathematica can properly determine these quantities for non-Hermitian matrices. They were recommended to refer to "The Algebraic Eigenvalue Problem" by J.H. Wilkinson for an example.
  • #1
physicsjock
89
0
Hey,

I have two quick questions,

Does mathematica automatically find the eigenvectors and values when you find the eigensystem of a non-Hermitian matrix?

I've been searching the net trying to find a way to find these vectors/values but everything I find briefly touches upon non-Hermitian without actually doing anything.

The method I have been trying is to apply the householder transformation and turn the matrix into a upper triangular hessian matrix, but after 10 applications of the householder transformation my matrix doesn't show any signs of change.

Could anyone point me in the right direction in finding the eigevectors and values of a non-Hermitian matrix?

Also if anyone knows if mathematica actually does properly determine these quantities of non-Hermitian matrix automatically when running eigensystem please let me know.

This line of code is the only part I'm not sure about, I have a feeling that mathematica doesn't determine the eigensystem of non-Hermitians properly.


Thanks in advanced
 
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  • #2
I've never actually seen an example of this before,

Try "They algebraic eigenvalue problem" by J.H.Wilkinson
 

FAQ: Finding Eigenvectors and Values of Non-Hermitian Matrices with Mathematica

What are Non-Hermitian matrices?

Non-Hermitian matrices are matrices that do not satisfy the properties of a Hermitian matrix. In other words, they are not equal to their conjugate transpose. This means that their eigenvalues may not be all real numbers and their eigenvectors may not be orthogonal.

What are the applications of Non-Hermitian matrices?

Non-Hermitian matrices have various applications in different fields such as quantum mechanics, statistical physics, and signal processing. They are used to model non-conservative systems, open quantum systems, and non-unitary dynamics.

How are Non-Hermitian matrices different from Hermitian matrices?

The main difference between Non-Hermitian and Hermitian matrices is that the latter satisfies the property of Hermiticity, which states that a matrix is equal to its conjugate transpose. This results in real eigenvalues and orthogonal eigenvectors. Non-Hermitian matrices do not have these properties, leading to complex eigenvalues and non-orthogonal eigenvectors.

Can Non-Hermitian matrices be diagonalized?

Yes, Non-Hermitian matrices can be diagonalized, but the resulting diagonal matrix may not have all real entries. This is in contrast to Hermitian matrices, which can always be diagonalized to a real diagonal matrix.

What are some techniques for dealing with Non-Hermitian matrices?

Some techniques for working with Non-Hermitian matrices include the Jordan decomposition, which breaks down the matrix into a diagonalizable part and a nilpotent part, and the biorthogonal system method, which involves finding the left and right eigenvectors of the matrix. Other methods include perturbation theory and numerical diagonalization algorithms.

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