Euclidean inner product

URI: https://gptkb.org/entity/Euclidean_inner_product

GPTKB entity

Predicate	Object
gptkbp:instanceOf	gptkb:mathematical_concept
gptkbp:alsoKnownAs	dot product
gptkbp:application	gptkb:machine_learning computer science engineering physics statistics
gptkbp:definedIn	gptkb:Euclidean_space
gptkbp:dimensions	finite-dimensional vector spaces
gptkbp:field	gptkb:geometry linear algebra
gptkbp:form	sum of products of corresponding components
gptkbp:generalizes	scalar product
gptkbp:hasSpecialCase	inner product
https://www.w3.org/2000/01/rdf-schema#label	Euclidean inner product
gptkbp:introduced	gptkb:Euclid
gptkbp:isBilinear	true
gptkbp:isPositiveDefinite	true
gptkbp:mapType	pair of vectors to a scalar
gptkbp:notation	x · y ⟨x, y⟩
gptkbp:property	⟨ax+by, z⟩ = a⟨x, z⟩ + b⟨y, z⟩ ⟨x, x⟩ = 0 iff x = 0 ⟨x, x⟩ ≥ 0 ⟨x, y⟩ = ⟨y, x⟩
gptkbp:relatedTo	gptkb:Cauchy-Schwarz_inequality gptkb:standard gptkb:Gram-Schmidt_process gptkb:Pythagorean_theorem orthonormal basis orthogonal projection angle formula
gptkbp:symmetry	true
gptkbp:usedFor	orthogonality measuring angle between vectors measuring length of vectors
gptkbp:bfsParent	gptkb:orthogonal_group
gptkbp:bfsLayer	5