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Orthonormality. Property of two or more vectors that are orthogonal and of unit length. In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors. A unit vector means that the vector has a length of 1, which is also known as normalized.
In certain cases, the word normal is used to mean orthogonal, particularly in the geometric sense as in the normal to a surface. For example, the y-axis is normal to the curve = at the origin. However, normal may also refer to the magnitude of a vector. In particular, a set is called orthonormal (orthogonal plus normal) if it is an orthogonal ...
Orthogonal matrix. In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is where QT is the transpose of Q and I is the identity matrix. This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to ...
Orthonormal basis. In mathematics, particularly linear algebra, an orthonormal basis for an inner product space with finite dimension is a basis for whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. [1][2][3] For example, the standard basis for a Euclidean space is an orthonormal basis, where the ...
In geometry, a normal is an object (e.g. a line, ray, or vector) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the line perpendicular to the tangent line to the curve at the point. A normal vector of length one is called a unit normal vector. A curvature vector is a normal vector ...
The first Frenet-Serret formula holds by the definition of the normal N and the curvature κ, and the third Frenet-Serret formula holds by the definition of the torsion τ. Thus what is needed is to show the second Frenet-Serret formula. Since T, N, B are orthogonal unit vectors with B = T × N, one also has T = N × B and N = B × T.
The concept of an orthogonal basis is applicable to a vector space (over any field) equipped with a symmetric bilinear form , where orthogonality of two vectors and means . For an orthogonal basis : where is a quadratic form associated with (in an inner product space, ). Hence for an orthogonal basis , where ...
Phrased differently: a matrix is normal if and only if its eigenspaces span C n and are pairwise orthogonal with respect to the standard inner product of C n. The spectral theorem for normal matrices is a special case of the more general Schur decomposition which holds for all square matrices.