eigenvalue

== English == === Etymology === From eigen- +‎ value. Partial calque of German Eigenwert. === Pronunciation === (Received Pronunciation) IPA(key): /ˈaɪ.ɡənˌvæl.juː/ (General American) IPA(key): /ˈaɪ.ɡənˌvæl.ju/ Hyphenation: ei‧gen‧val‧ue === Noun === eigenvalue (plural eigenvalues) (linear algebra) A scalar λ {\displaystyle \lambda } , such that there exists a non-zero vector x {\displaystyle \mathrm {x} } (a corresponding eigenvector) for which the image of x {\displaystyle \mathrm {x} } under a given linear transformation A {\displaystyle A} is equal to the image of x {\displaystyle \mathrm {x} } under multiplication by λ {\displaystyle \lambda } ; i.e. A x = λ x {\displaystyle A\mathrm {x} =\lambda \mathrm {x} } . Synonyms: characteristic root, characteristic value, eigenroot, latent value, proper value Hyponyms: left eigenvalue, right eigenvalue ==== Usage notes ==== Eigenvalue is the standard term in English, displacing the older proper value. When unqualified, eigenvalue conventionally refers to a right eigenvalue, characterised by M x = λ x {\displaystyle M\mathrm {x} =\lambda \mathrm {x} } for an associated right eigenvector x {\displaystyle \mathrm {x} } . Left eigenvalues, characterised by y M = y λ {\displaystyle \mathrm {y} M=\mathrm {y} \lambda } also exist with an associated left eigenvectors y {\displaystyle \mathrm {y} } . ==== Derived terms ==== ==== Translations ====