# Approximate diagonalisation of (3,3) hermitian matrix

• SUSY
In summary, the speaker has a complicated 3x3 hermitian matrix K that they need to diagonalize by finding a unitary matrix S such that S^{\dagger} K S is diagonal. They are wondering if it is possible to find approximate expressions for the entries of S in terms of the entries in K, and if it is possible to decompose S into matrices that can be approximated with such expressions. They mention that the entries in K are given as a power series and they are only interested in the lowest non-vanishing order. They ask if anyone knows of a method to do this.
SUSY
Hi,

I have a 3 by 3 hermitian matrix K that I need to diagonalise. More accurately, I am searching for a unitary matrix S such that $$S^{\dagger} K S$$ is diagonal.

The problem is that K is very complicated and the expression for S in mathematica takes up quiiiiiet a lot of space.

Is it possible to find approximate expressions for the entries of S in terms of entries in K? What I am looking for is something like (as an example) $$S_{ij}=K_{ij} + K_{ji} bla bla$$
Or is it maybe possible to decompose S into matrices which I can then approximate by such expressions (they can, of course, be more complicated than the example given above).

Since the entries in K are given as a power series of some small parameter ε $$K_{ij}=a_0 + a_1 \epsilon + ...$$ and I am only interested in the lowest non-vanishing order anyway, it would be nice to have expressions for the entries in S in terms of the entries in K. Then, I could easily evaluate the entries in S to lowest non-vanishing order in ε (so I am only interested in an approximate S anyway).

Does anybody know of such a method?

Thanks,
Susy

If you can write ##K = K_0 + \epsilon K_\epsilon##, can you diagonalize ##K_0## and ##K_\epsilon## simultaneously? This is routinely done numerically, as a generalized eigenproblem, but I don't know about doing it symbolically.

Yes, you should be able to diagonalize both simultaneously. In all cases I can think of, $K_0$ will already be a diagonal matrix and $K_{\epsilon}$ is a correction ($K_{\epsilon}$ can in principle be an arbitrary hermitian matrix).

## 1. What is approximate diagonalisation of a (3,3) Hermitian matrix?

Approximate diagonalisation is a mathematical technique used to simplify the calculation of eigenvalues and eigenvectors of a Hermitian matrix. It involves approximating the matrix with a simpler, diagonal matrix that has the same eigenvalues as the original matrix. This technique is particularly useful when dealing with large matrices, as it reduces the computational complexity of finding eigenvalues and eigenvectors.

## 2. Why is the (3,3) Hermitian matrix important?

The (3,3) Hermitian matrix is important because it is a specific type of square matrix that has several useful properties. It is symmetric, meaning that its transpose is equal to itself, and it has only real eigenvalues. These properties make it useful in a wide range of applications, including quantum mechanics and signal processing.

## 3. How is approximate diagonalisation different from exact diagonalisation?

Exact diagonalisation involves finding the exact eigenvalues and eigenvectors of a matrix, while approximate diagonalisation provides an approximation of these values. Approximate diagonalisation is typically used when exact diagonalisation is not feasible due to the size or complexity of the matrix.

## 4. What are the benefits of using approximate diagonalisation?

The main benefit of approximate diagonalisation is that it reduces the computational complexity of finding eigenvalues and eigenvectors of a matrix. This can be particularly useful when dealing with large matrices, as it speeds up the calculation process. Additionally, approximate diagonalisation can also improve the accuracy of the solutions when compared to other approximation techniques.

## 5. Are there any limitations to using approximate diagonalisation?

One limitation of approximate diagonalisation is that it can only be used for Hermitian matrices, which have real eigenvalues. It also provides an approximation of the eigenvalues and eigenvectors, rather than exact values, which may not be suitable for certain applications. Additionally, the accuracy of the solutions may be affected by the quality of the approximation used.

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