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The vector projection of a vector on a nonzero vector is defined as [note 1] = , , , where , denotes the inner product of the vectors and . This means that proj u ( v ) {\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )} is the orthogonal projection of v {\displaystyle \mathbf {v} } onto the line spanned by u ...
A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the vectors are all perpendicular to each other. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis.
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 Q T Q = Q Q T = I , {\displaystyle Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,} where Q T is the transpose of Q and I is the identity matrix .
[1] [2] [3] For example, the standard basis for a Euclidean space is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation ) is also orthonormal, and every orthonormal basis for R n {\displaystyle \mathbb {R} ^{n ...
The Generalized vector space model is a generalization of the vector space model used in information retrieval. Wong et al. [1] presented an analysis of the problems that the pairwise orthogonality assumption of the vector space model (VSM) creates. From here they extended the VSM to the generalized vector space model (GVSM).
In numerical linear algebra, the Arnoldi iteration is an eigenvalue algorithm and an important example of an iterative method.Arnoldi finds an approximation to the eigenvalues and eigenvectors of general (possibly non-Hermitian) matrices by constructing an orthonormal basis of the Krylov subspace, which makes it particularly useful when dealing with large sparse matrices.
There are a number of matrix norms that act on the singular values of the matrix. Frequently used examples include the Schatten p-norms, with p = 1 or 2. For example, matrix regularization with a Schatten 1-norm, also called the nuclear norm, can be used to enforce sparsity in the spectrum of a matrix.
PGF/TikZ is a pair of languages for producing vector graphics (e.g., technical illustrations and drawings) from a geometric/algebraic description, with standard features including the drawing of points, lines, arrows, paths, circles, ellipses and polygons.