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41 relations: Cholesky decomposition, Condition number, Determinant, Eigendecomposition of a matrix, Eigenvalue algorithm, Floating-point arithmetic, Givens rotation, Gram–Schmidt process, Householder transformation, Hyperplane, Inner product space, Invertible matrix, Iwasawa decomposition, Linear algebra, Linear independence, Linear least squares (mathematics), Linear span, Loss of significance, LU decomposition, Matrix (mathematics), Matrix decomposition, Minor (linear algebra), Numerical stability, Orthogonal matrix, Orthogonality, Orthonormal basis, Permutation matrix, Plane (geometry), Polar decomposition, QR algorithm, Rank (linear algebra), Row and column spaces, RRQR factorization, Singular-value decomposition, Spectral theorem, Square matrix, Triangle, Triangular matrix, Unit vector, Unitary matrix, Vector projection.
Cholesky decomposition
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced /ʃ-/) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g. Monte Carlo simulations.
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Condition number
In the field of numerical analysis, the condition number of a function with respect to an argument measures how much the output value of the function can change for a small change in the input argument.
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Determinant
In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.
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Eigendecomposition of a matrix
In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors.
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Eigenvalue algorithm
In numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix.
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Floating-point arithmetic
In computing, floating-point arithmetic is arithmetic using formulaic representation of real numbers as an approximation so as to support a trade-off between range and precision.
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Givens rotation
In numerical linear algebra, a Givens rotation is a rotation in the plane spanned by two coordinates axes.
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Gram–Schmidt process
In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalising a set of vectors in an inner product space, most commonly the Euclidean space Rn equipped with the standard inner product.
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Householder transformation
In linear algebra, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane containing the origin.
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Hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space.
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Inner product space
In linear algebra, an inner product space is a vector space with an additional structure called an inner product.
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Invertible matrix
In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate) if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.
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Iwasawa decomposition
In mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (a consequence of Gram–Schmidt orthogonalization).
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Linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.
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Linear independence
In the theory of vector spaces, a set of vectors is said to be if one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be.
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Linear least squares (mathematics)
In statistics and mathematics, linear least squares is an approach to fitting a mathematical or statistical model to data in cases where the idealized value provided by the model for any data point is expressed linearly in terms of the unknown parameters of the model.
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Linear span
In linear algebra, the linear span (also called the linear hull or just span) of a set of vectors in a vector space is the intersection of all subspaces containing that set.
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Loss of significance
Loss of significance is an undesirable effect in calculations using finite-precision arithmetic such as floating-point arithmetic.
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LU decomposition
In numerical analysis and linear algebra, LU decomposition (where "LU" stands for "lower–upper", and also called LU factorization) factors a matrix as the product of a lower triangular matrix and an upper triangular matrix.
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Matrix (mathematics)
In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
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Matrix decomposition
In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices.
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Minor (linear algebra)
In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns.
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Numerical stability
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms.
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Orthogonal matrix
In linear algebra, an orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e. where I is the identity matrix.
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Orthogonality
In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms.
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Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other.
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Permutation matrix
\pi.
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Plane (geometry)
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.
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Polar decomposition
In mathematics, particularly in linear algebra and functional analysis, the polar decomposition of a matrix or linear operator is a factorization analogous to the polar form of a nonzero complex number z as z.
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QR algorithm
In numerical linear algebra, the QR algorithm is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix.
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Rank (linear algebra)
In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns.
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Row and column spaces
In linear algebra, the column space (also called the range or '''image''') of a matrix A is the span (set of all possible linear combinations) of its column vectors.
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RRQR factorization
An RRQR factorization or rank-revealing QR factorization is a matrix decomposition algorithm based on the QR factorization which can be used to determine the rank of a matrix.
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Singular-value decomposition
In linear algebra, the singular-value decomposition (SVD) is a factorization of a real or complex matrix.
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Spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis).
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Square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns.
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Triangle
A triangle is a polygon with three edges and three vertices.
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Triangular matrix
In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix.
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Unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1.
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Unitary matrix
In mathematics, a complex square matrix is unitary if its conjugate transpose is also its inverse—that is, if where is the identity matrix.
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Vector projection
The vector projection of a vector a on (or onto) a nonzero vector b (also known as the vector component or vector resolution of a in the direction of b) is the orthogonal projection of a onto a straight line parallel to b. It is a vector parallel to b, defined as where a_1 is a scalar, called the scalar projection of a onto b, and b̂ is the unit vector in the direction of b. In turn, the scalar projection is defined as where the operator · denotes a dot product, |a| is the length of a, and θ is the angle between a and b. The scalar projection is equal to the length of the vector projection, with a minus sign if the direction of the projection is opposite to the direction of b. The vector component or vector resolute of a perpendicular to b, sometimes also called the vector rejection of a from b, is the orthogonal projection of a onto the plane (or, in general, hyperplane) orthogonal to b. Both the projection a1 and rejection a2 of a vector a are vectors, and their sum is equal to a, which implies that the rejection is given by.
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References
[1] https://en.wikipedia.org/wiki/QR_decomposition
