eigenvalue
UK[ˈaɪɡənˌvæljuː]US[ˈaɪɡənˌvæljuː]
n
a special number associated with a linear transformation of a vector space, which when multiplied by a corresponding eigenvector yields the same result as the transformation applied to that eigenvector, only scaled by that number.
n
(in general use) a characteristic value or factor, especially one that is quantitatively significant.
Morpheme Breakdown
eigen
value
eigen
own, characteristic
value
numerical worth
Etymology
the term is a partial calque from the german word 'eigenwert', literally meaning 'own value' or 'characteristic value'. it was coined in the early 20th century within the context of german mathematical physics, particularly in the work of david hilbert on integral equations. the german 'eigen' was retained in its original form due to its precise technical meaning, while 'wert' was translated into the english 'value'. this hybrid construction elegantly captures the concept of an intrinsic, unchanging property (the 'own' part) of a mathematical operator that manifests as a specific numerical quantity (the 'value').
Analysis
Structure: eigen (own, characteristic) + value (numerical worth)
'eigen' is a germanic (specifically german) morpheme meaning 'own', 'inherent', or 'characteristic'. 'value' is from old french/latin, meaning 'worth' or 'amount'.
Examples
the dominant eigenvalue of the matrix determines the system's long-term behavior.
in quantum mechanics, the possible results of a measurement are the eigenvalues of the corresponding operator.
the stability of the model is analyzed by calculating the eigenvalues of its jacobian matrix.