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A simple, randomized algorithm for diagonalizing normal matrices
1Institute of Mathematics, École Polytechnique Fédérale de Lausanne (EPFL), 1015, Lausanne, Switzerland.
Calcolo
|July 29, 2025
Summary
A new numerical method simplifies complex normal matrix diagonalization. It achieves this by transforming the matrix into a Hermitian form using random linear combinations.
Area of Science:
- Numerical analysis
- Linear algebra
- Matrix theory
Background:
- Complex normal matrices are a class of matrices with applications in quantum mechanics and signal processing.
- Diagonalization is a fundamental operation for simplifying matrix analysis and solving systems of equations.
Purpose of the Study:
- To introduce and analyze a novel, simple numerical method for diagonalizing complex normal matrices.
- To demonstrate the efficacy of the proposed method in simplifying matrix computations.
Main Methods:
- The method involves constructing a Hermitian matrix from a random linear combination of the Hermitian and skew-Hermitian parts of the original complex normal matrix.
- The core technique relies on the properties of random matrix theory and spectral decomposition.
Main Results:
- The proposed numerical method successfully diagonalizes complex normal matrices.
- Analysis confirms the method's simplicity and efficiency compared to existing techniques.
Conclusions:
- This simple numerical method offers an effective approach for diagonalizing complex normal matrices.
- The technique provides a valuable tool for researchers and practitioners in fields utilizing complex normal matrix analysis.
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