4 The Standard Eigenvalue Problem

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Problem 4: The Standard Eigenvalue Problem

Let $n \in N$ . Let $A \in R^{n \times n}$ be a given square matrix. The eigenvalue problem is to find all scalars $λ$ and all possible nonzero $n \times 1$ vectors $\vec{x}$ such that

A \cdot \vec{x} = λ \vec{x}

This is a powerful approach to use when analyzing coupled harmonic oscillators. The standard eigenvalue problem is essential to many topics in physical sciences and is the backbone of the entire field of vibrational mechanics. It is useful to think about the eigenvalue problems as describing and encoding a special class of differential equations.

Examples of modeling problems that involve differential equations include the analysis of the displacements of masses in a mass-spring chain, RCL circuit analysis, the solution to partial differential equations, analysis of vibrations, facial recognition software, and principal component analysis.