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The 69 pages that link to Eigenvalue, each with the reason it gives.

• DeterminantRelated: Their product, counted with algebraic multiplicity, equals the determinant.

• Control theoryRelated: System eigenvalues determine the modes and stability of linear dynamics.

• Linear algebraRelated: Eigenvalues expose directions that a transformation stretches without changing direction.

• Quantum numberRelated: Measured values of quantized observables are eigenvalues, often used as quantum numbers.

• Principal component analysisRelated: In covariance-based PCA, each eigenvalue measures the variance captured by its component.

• Rotation matrixRelated: Eigenvalues reveal invariant directions and rotation angles in special cases.

• Stability theoryRelated: Linear stability tests use eigenvalue real parts to identify growing or decaying modes.

• Condition numberRelated: Eigenvalue condition numbers quantify sensitivity of eigenvalues to matrix perturbations.

• Hessian matrixRelated: Hessian eigenvalues indicate curvature along principal directions.

• Singular value decompositionRelated: Squared singular values are eigenvalues of the matrix’s associated Gram matrices.

• Characteristic polynomialRelated: Its values are exactly the roots of the characteristic polynomial.

• ObservableRelated: Eigenvalues supply the possible outcomes in measurements of discrete-spectrum observables.

• Angular momentum operatorRelated: Angular-momentum measurements yield eigenvalues of the corresponding component or squared operator.

• Jordan normal formRelated: Each Jordan block places one eigenvalue along its diagonal.

• Linear operatorRelated: Eigenvalues quantify the operator's action along special directions.

• Polynomial rootRelated: Eigenvalues are not polynomial roots themselves, but they are roots of the characteristic polynomial.

• Factor analysisRelated: Eigenvalues help summarize how much variance candidate factors represent.

• Canonical commutation relationRelated: Eigenstates of position or momentum expose why the two observables cannot share a complete set of sharp values.

• EigenvectorRelated: It gives the scale factor paired with each preserved direction.

• Orthogonal matrixRelated: Orthogonal matrices have eigenvalues of absolute value one, including possible complex eigenvalues.

• Bifurcation theoryRelated: Eigenvalues indicate local stability and identify several bifurcation thresholds.

• Matrix mechanicsRelated: Observable eigenvalues correspond to the possible results of ideal measurements.

• Square matrixRelated: Eigenvalues are defined for square matrices, unlike singular values, which also apply to rectangular ones.

• BifurcationRelated: Eigenvalues of a system’s linearization help identify local stability changes.

• Cayley–Hamilton theoremRelated: Eigenvalues are roots of the characteristic polynomial, though the theorem is stronger than its eigenvalue consequences.

• Compact operatorRelated: Every nonzero spectral value of a compact operator is an eigenvalue.

• Matrix inverseRelated: A square matrix is invertible exactly when zero is not one of its eigenvalues.

• Symmetric matrixRelated: Real symmetric matrices have only real eigenvalues.

• Floquet theoryRelated: Eigenvalues of the one-period evolution map become the Floquet multipliers.

• Pauli matricesRelated: The eigenvalues ±1 yield spin-component outcomes of ±ħ/2 after the physical scaling.

• Positive-definite matrixRelated: A real symmetric matrix is positive definite exactly when every eigenvalue is positive.

• Wave mechanicsRelated: Allowed energy levels arise as eigenvalues of the Hamiltonian.

• Diagonal matrixRelated: Each diagonal entry is an eigenvalue, with its coordinate direction as an eigenvector.

• Particle in a boxRelated: Allowed energies arise as eigenvalues of the Hamiltonian.

• Saddle pointRelated: Eigenvalue signs or real parts classify stability along local directions.

• Jordan decompositionRelated: Eigenvalues determine the semisimple action and the generalized eigenspaces.

• Routh–Hurwitz stability criterionRelated: Eigenvalues of a system matrix are roots of its characteristic polynomial.

• Schur's lemmaRelated: Over an algebraically closed field, an endomorphism has an eigenvalue whose subtraction yields a noninvertible intertwiner.

• Zero of a functionRelated: Eigenvalues are zeros of the characteristic polynomial of a matrix.

• Position operatorRelated: Position measurements correspond to eigenvalues of the position operator.

• Newton's identitiesRelated: Their power sums and elementary symmetric functions satisfy Newton's identities.

• Sylvester's law of inertiaRelated: The signs of the eigenvalues give the positive and negative counts in an orthogonal diagonalization.

• Crystallographic restriction theoremRelated: Rotation eigenvalues encode the angle whose allowed orders the theorem limits.

• Euler's rotation theoremRelated: Every three-dimensional proper rotation has eigenvalue one, yielding an unchanged axis direction.

• QuantizationRelated: Possible outcomes of measuring an observable are its operator’s eigenvalues.

• Sylvester's criterionRelated: A Hermitian matrix is positive definite exactly when all its eigenvalues are positive.

• Triangular matrixRelated: The eigenvalues of a triangular matrix are its diagonal entries, counted with multiplicity.

• Jordan–Chevalley decompositionRelated: The semisimple component preserves the operator’s eigenvalues.

• Selberg trace formulaRelated: Laplacian eigenvalues form the discrete spectral data on the formula’s left side.

• Sign (mathematics)Related: An eigenvalue's sign can indicate whether its eigendirection is preserved or reversed.