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In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity to the linear algebra of bilinear forms. Two elements u and v of a vector space with bilinear form B {\displaystyle B} are orthogonal when B ( u , v ) = 0 {\displaystyle B(\mathbf {u} ,\mathbf {v} )=0} .
In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form.When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the interval:
The line segments AB and CD are orthogonal to each other. In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity.Whereas perpendicular is typically followed by to when relating two lines to one another (e.g., "line A is perpendicular to line B"), [1] orthogonal is commonly used without to (e.g., "orthogonal lines A and B").
The orthogonality and completeness of this set of solutions follows at once from the larger framework of Sturm–Liouville theory. The differential equation admits another, non-polynomial solution, the Legendre functions of the second kind. A two-parameter generalization of (Eq.
It induces a notion of orthogonality in the usual way, namely that two polynomials are orthogonal if their inner product is zero. Then the sequence ( P n ) ∞ n =0 of orthogonal polynomials is defined by the relations deg P n = n , P m , P n = 0 for m ≠ n . {\displaystyle \deg P_{n}=n~,\quad \langle P_{m},\,P_{n}\rangle =0\quad {\text ...
More accurately, the general orthogonality principle states the following: Given a closed subspace of estimators within a Hilbert space and an element in , an element ^ achieves minimum MSE among all elements in if and only if {(^)} = for all .
Here orthogonality is important not only for reducing A T A = (R T Q T)QR to R T R, but also for allowing solution without magnifying numerical problems. In the case of a linear system which is underdetermined, or an otherwise non-invertible matrix, singular value decomposition (SVD) is equally useful.
The construction of orthogonality of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces. In the Cartesian plane, two vectors are said to be perpendicular if the angle between them is 90° (i.e. if they form a right angle).