How can I analyse and classify a matrix?

[SeM]

Hi, I have a matrix of an ODE which yields complex eigenvalues and eigenvectors. It is therefore not Hermitian. How can I further analyse the properties of the matrix in a Hilbert space?
The idea is to reveal the properties of stability and instability of the matrix. D_2 and D_1 are the second and first order derivatives respectively, and a and b are real numbers.
I thought of treating the two operators as x² and x and solve the quadratic equation, however, this does not really give much more information.

 

[FactChecker]

The stability properties are indicated by the position of the eigenvalues on the complex plane. A modulus greater than 1 indicates exponential growth when applied to the associated eigenvector. If an eigenvalue is complex, its argument indicates a rotation and a cyclic behavior.

 

[SeM]

Thanks. Does this imply that there is no more classification I can actually do, given that the eigenvalue is complex, and the Matrix appears as neither hermitian, unitary, norm or Skew Hermitian?

 

[FactChecker]

It sounds like you are looking for a name to describe your matrix. I'm not aware of any others to try.

If you want to describe the nature of the matrix, you should describe it in terms of the individual eigenvalues and associated eigenvectors. You can describe the dynamic modes of the system. You could examine if the matrix is separable (see https://en.wikipedia.org/wiki/Singular-value_decomposition ). You can also check if the matrix is ill-conditioned. If it was normal, the condition number would be the ratio of the modulus of the largest and smallest eigenvalues. But since it is not normal, I am not sure how to determine the condition number. (see https://en.wikipedia.org/wiki/Condition_number )

 

[SeM]

Thanks Fastchecker, this was very clear. I have tried to derive the eigenvalues, but given that the matrix contains two numbers and two operators, hence like:

\begin{bmatrix}
D_4 & D_1 \\
i5 & 6 \\
\end{bmatrix}

where D4 is the fourth derivative and D1 is the first derivative, I end up with a secular solution but not based on numbers, but with the operators in the roots. Is this the regular procedure to solve a secular equation with operators and numbers in the elements and if it is, how can one use this result to say something about the matrix? I have only worked with numbers in the elements before, so I am no sure here. Sorry!

In this case there are parts of the secular equation solution which look like:

D4 (x_1x_2), so the fourth derivative of the two eigenvector coordinates multiplied by one another. Does one treat that as D4x^2 , which is 0?

Thanks

 

[StoneTemplePython]

...(see https://en.wikipedia.org/wiki/Singular-value_decomposition ). You can also check if the matrix is ill-conditioned. If it was normal, the condition number would be the ratio of the modulus of the largest and smallest eigenvalues. But since it is not normal, I am not sure how to determine the condition number. (see https://en.wikipedia.org/wiki/Condition_number )

You mentioned SVD -- just need to make the connection here. As long as we're using an L2/ euclidean norm for measuring variations, use singular values to measure condition number. In the special case of a normal matrix, your singular values map exactly to magnitudes of eigenvalues. In general for square matrices, your largest singular value is always $\geq$ magnitude of largest eigenvalue and smallest singular values is always $\leq$ smallest eigenvalues magnitude. (why?)