How many orthonormal eigenvector bases does a symmetric n ×n...

Key Concepts

Spectral Theorem

The Spectral Theorem is a fundamental result stating that any real symmetric matrix can be diagonalized by an orthogonal matrix. This means that the matrix has a complete set of eigenvectors which can be chosen to be orthonormal, and the matrix can be written in a simplified, diagonal form using its eigenvalues. This concept underpins the idea of expressing matrices in terms of their eigenvectors and eigenvalues.

Orthogonal Diagonalization

Orthogonal diagonalization refers to the process of decomposing a symmetric matrix into an orthogonal matrix and a diagonal matrix, where the columns of the orthogonal matrix are eigenvectors of the original matrix. This process highlights that the set of eigenvectors can be arranged into an orthonormal basis, which is essential for many applications in linear algebra and numerical analysis.

Eigenvalue Multiplicity

Eigenvalue multiplicity plays a critical role in determining the variability of the eigenvector basis. When eigenvalues are distinct, the corresponding eigenvectors are determined up to a sign (and permutation), but when an eigenvalue has a higher multiplicity, its eigenspace is of higher dimension and there are infinitely many ways to choose an orthonormal basis for that subspace. This intrinsic property is key to answering how many orthonormal eigenvector bases exist for a given symmetric matrix.

Eigenvector Ambiguity

There is inherent ambiguity in the choice of eigenvectors even when the eigenvalues are distinct. Each eigenvector is unique only up to a scalar multiple, and when normalized, up to a sign change. Additionally, the order of the eigenvectors in the basis is arbitrary. This ambiguity, combined with the freedom to choose any orthonormal basis within an eigenspace corresponding to a repeated eigenvalue, explains why there can be infinitely many orthonormal eigenvector bases for a symmetric matrix.

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