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6: Eigenvalue Problems

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  • An eigenvalue problem is a matrix equation of the form

    \[\mathbf{A} \vec{x} = \lambda \vec{x},\]

    where \(\mathbf{A}\) is a known \(N\times N\) matrix. The problem is to find one (or more than one) non-zero vector \(\vec{x}\), which is called an eigenvector, and the associated \(\lambda \in \mathbb{C}\), which is called an eigenvalue. Eigenvalue problems are ubiquitous in practically all fields of physics. Most prominently, they are used to describe the "modes" of a physical system, such as the modes of a classical mechanical oscillator, or the energy states of an atom.

    Before discussing numerical solutions to the eigenvalue problem, let us quickly review the relevant mathematical facts.

    • 6.1: Basic Facts about Eigenvalue Problems
    • 6.2: Numerical Eigensolvers
      There exist numerical methods, called eigensolvers, which can compute eigenvalues (and eigenvectors) even for very large matrices, with hundreds of rows/columns, or larger. How could this be? The answer is that numerical eigensolvers are approximate, not exact. But even though their results are not exact, they are very precise—they can approach the exact eigenvalues to within the fundamental precision limits of floating-point arithmetic.

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