A basis of a vector space composed of eigenvectors of a linear operator so that the operator acts diagonally on coordinate representations relative to that basis.

Principle

If a linear operator is diagonalizable, there exists a complete set of linearly independent eigenvectors whose span equals the space; for normal self-adjoint operators one can choose an orthonormal eigenbasis (spectral theorem).

Demonstration

The Laplace operator on the circle has an eigenbasis of complex exponentials e^{ikθ}; expressing a function in that basis diagonalizes the Laplacian and converts PDEs into ordinary algebraic relations on coefficients.

Misapplication

Assuming an arbitrary non-diagonalizable matrix has an eigenbasis and attempting to diagonalize it without accounting for Jordan chains and generalized eigenvectors.

Consequence

An eigenbasis reduces linear dynamics to independent scalar evolutions along each eigenvector, simplifies spectral decompositions, and facilitates efficient computations such as modal truncation and diagonal exponentiation.

Reversal

In the absence of a full eigenbasis one uses a Jordan basis of generalized eigenvectors or a singular value decomposition that provides orthogonal modes for non-normal operators but not true eigenvectors for diagonalization.

Boundary

Existence depends on the operator and underlying field: finite-dimensional diagonalizability requires algebraic multiplicities equal geometric multiplicities; in infinite dimensions completeness and orthogonality require additional spectral conditions.

Semantic Tension

Confused with singular vectors from SVD: singular vectors diagonalize a related positive operator and are orthogonal for arbitrary matrices, whereas eigenvectors diagonalize the original operator but may not be orthogonal if the operator is non-normal.

Synthesis

An eigenbasis is a complete set of eigenvectors furnishing coordinates in which a diagonalizable linear operator acts by scalar multiplication, enabling decomposition of dynamics and operators into independent modal contributions.