Hermitian forms (work on quadratic forms)
E502192
Hermitian forms (work on quadratic forms) are a class of complex-valued quadratic forms that are linear in one variable and conjugate-linear in the other, generalizing real symmetric quadratic forms and playing a central role in linear algebra and functional analysis.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Hermitian form | 0 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
quadratic form ⓘ sesquilinear form ⓘ |
| additivityCondition | h(x+y,z) = h(x,z) + h(y,z) ⓘ |
| appliesTo |
finite-dimensional complex vector spaces
ⓘ
infinite-dimensional Hilbert spaces ⓘ |
| associatedQuadraticForm | q(x) = h(x,x) ⓘ |
| associatedQuadraticFormProperty | q(x) is real-valued ⓘ |
| classificationInvariants |
nullity
ⓘ
rank ⓘ signature in the indefinite case ⓘ |
| conjugateHomogeneityCondition | h(x,αy) = overline(α) h(x,y) ⓘ |
| conjugateSymmetryCondition | h(x,y) = overline(h(y,x)) ⓘ |
| definedOver | complex vector space ⓘ |
| determines | Hermitian matrix ⓘ |
| diagonalizationProperty | Hermitian matrix can be unitarily diagonalized ⓘ |
| eigenvalueProperty | associated Hermitian matrix has real eigenvalues ⓘ |
| fieldOfStudy |
complex geometry
ⓘ
functional analysis ⓘ linear algebra ⓘ operator theory ⓘ |
| generalizes |
real quadratic form
ⓘ
real symmetric bilinear form ⓘ |
| hasProperty |
conjugate symmetric
ⓘ
conjugate-linear in second argument ⓘ linear in first argument ⓘ |
| hasVariant |
degenerate Hermitian form
ⓘ
indefinite Hermitian form ⓘ positive definite Hermitian form ⓘ |
| historicalOrigin | generalization of symmetric bilinear forms to complex fields ⓘ |
| homogeneityCondition | h(αx,y) = α h(x,y) ⓘ |
| invariantUnder | unitary change of basis ⓘ |
| nameOrigin | named after Charles Hermite ⓘ |
| nondegeneracyCondition | h(x,y)=0 for all y implies x=0 ⓘ |
| orthogonalityCondition | x ⟂ y iff h(x,y)=0 ⓘ |
| relatedConcept |
Hermitian operator
ⓘ
complex inner product ⓘ sesquilinear form ⓘ unitary operator ⓘ |
| representedBy | Hermitian matrix with respect to a basis ⓘ |
| specialCase | inner product when positive definite ⓘ |
| usedFor |
classification of complex quadratic forms
ⓘ
defining inner product on complex vector spaces ⓘ defining norms on complex vector spaces ⓘ defining orthogonality ⓘ geometry of complex projective spaces ⓘ spectral theory of normal operators ⓘ unitary diagonalization of matrices ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.