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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").
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} .
Toggle Orthogonality principle for linear estimators subsection. 1.1 Example. ... Since the principle is a necessary and sufficient condition for optimality, ...
The condition Q T Q = I says that the columns of Q are orthonormal. This can only happen if Q is an m × n matrix with n ≤ m (due to linear dependence). Similarly, QQ T = I says that the rows of Q are orthonormal, which requires n ≥ m .
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.
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 associated Legendre polynomials are not mutually orthogonal in general. For example, is not orthogonal to .However, some subsets are orthogonal. Assuming 0 ≤ m ≤ ℓ, they satisfy the orthogonality condition for fixed m:
The expansion coefficients are the analogs of Fourier coefficients, and can be obtained by multiplying the above equation by the complex conjugate of a spherical harmonic, integrating over the solid angle Ω, and utilizing the above orthogonality relationships. This is justified rigorously by basic Hilbert space theory.