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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").
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.
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:
Because this equation holds for all vectors, p, one concludes that every rotation matrix, Q, satisfies the orthogonality condition, Q T Q = I . {\displaystyle Q^{\mathsf {T}}Q=I.} Rotations preserve handedness because they cannot change the ordering of the axes, which implies the special matrix condition,
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 ...
Let (()) = be a sequence of orthogonal polynomials defined on the interval [,] satisfying the orthogonality condition () =,, where () is a suitable weight function, is a constant depending on , and , is the Kronecker delta.