The Bauer-Fike theorem, published by Friedrich L. Bauer and C. T. Fike in 1960, is a result in matrix perturbation theory giving an absolute upper bound on how far an eigenvalue of a perturbed diagonalizable matrix can deviate from the nearest eigenvalue of the original, unperturbed matrix. The bound is expressed through the condition number of the matrix of eigenvectors, so that a matrix whose eigenvectors are close to being linearly dependent has eigenvalues that are correspondingly more sensitive to perturbation. The theorem is a standard tool in numerical analysis for judging how reliable a computed eigenvalue is once rounding error or other small perturbations are taken into account.
Facts
StatementFor any eigenvalue of a perturbed diagonalizable matrix, there exists an eigenvalue of the original matrix such that the difference between them is bounded by the condition number of the matrix of eigenvectors multiplied by the size of the perturbation. 2 Classification
Statement Form Connections
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. Wikipedia: Bauer-Fike theorem
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
In its substance, it states an absolute upper bound for the deviation of one perturbed matrix eigenvalue from a properly chosen eigenvalue of the exact matrix.
2. Bauer-Fike theorem, Wikipedia
Lede section
In mathematics, the Bauer-Fike theorem is a standard result in the perturbation theory of the eigenvalue of a complex-valued diagonalizable matrix.
History section
The theorem was proved by Friedrich L. Bauer and C. T. Fike in 1960.
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