Spectral theorem

The spectral theorem states that operators in specified classes, including self-adjoint operators on finite-dimensional inner-product spaces, have orthonormal bases of eigenvectors. It also characterizes these operators through diagonal or spectral decompositions.

Eigenvalue: A scalar by which a linear transformation scales a nonzero vector without changing its direction. Each basis vector in the finite-dimensional theorem is scaled by its eigenvalue.
Inner product space: A vector space equipped with an inner product that defines lengths and angles. Orthonormal eigenvectors are defined using its geometry.
Jordan normal form: A canonical matrix form built from eigenvalues and blocks that encode generalized eigenvectors. It handles matrices that lack a full eigenbasis, unlike the diagonalizable cases of the theorem.
Quantum observable: A self-adjoint operator representing a measurable physical quantity in quantum mechanics. Its spectral projections encode possible measurement outcomes and their probabilities.
Eigenvector: A nonzero vector whose image under a linear transformation is a scalar multiple of itself. The theorem assembles eigenvectors into an orthonormal basis.
Self-adjoint operator: An operator equal to its adjoint on a specified inner-product space and domain. Self-adjointness is the central hypothesis in the standard spectral theorem.
Diagonalizable matrix: A square matrix similar to a diagonal matrix over a specified field. Diagonalizability needs an eigenbasis but does not require it to be orthonormal.
Principal component analysis: A statistical method that finds orthogonal directions of greatest variance in data. It uses eigenvectors of a covariance matrix to identify principal directions.
Orthogonal diagonalization: A matrix factorization expressing a real symmetric matrix as an orthogonal change of basis of a diagonal matrix. It is the matrix form of the real finite-dimensional spectral theorem.
Normal operator: An operator that commutes with its adjoint. Finite-dimensional complex normal operators also have orthonormal eigenbases.

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