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Matrix Determinant Lemma
In mathematics, in particular linear algebra, the matrix determinant lemma computes the determinant of the sum of an invertible matrix A and the dyadic product, uvT, of a column vector u and a row vector vT. Statement Suppose A is an invertible square matrix and u, v are column vectors. Then the matrix determinant lemma states that :\det\left(\mathbf + \mathbf^\textsf\right) = \left(1 + \mathbf^\textsf\mathbf^\mathbf\right)\,\det\left(\mathbf\right)\,. Here, uvT is the outer product of two vectors u and v. The theorem can also be stated in terms of the adjugate matrix of A: :\det\left(\mathbf + \mathbf^\textsf\right) = \det\left(\mathbf\right) + \mathbf^\textsf\mathrm\left(\mathbf\right)\mathbf\,, in which case it applies whether or not the square matrix A is invertible. Proof First the proof of the special case A = I follows from the equality: : \begin \mathbf & 0 \\ \mathbf^\textsf & 1 \end \begin \mathbf + \mathbf^\textsf & \mathbf \\ 0 & 1 \end \begin \mathbf & 0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Outer Product
In linear algebra, the outer product of two coordinate vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensiona ...s is a Matrix (mathematics), matrix. If the two vectors have dimensions ''n'' and ''m'', then their outer product is an ''n'' × ''m'' matrix. More generally, given two tensors (multidimensional arrays of numbers), their outer product is a tensor. The outer product of tensors is also referred to as their tensor product, and can be used to define the tensor algebra. The outer product contrasts with: * The dot product (a special case of "inner product"), which takes a pair of coordinate vectors as input and produces a Scalar (mathematics), scalar * The Kronecker product, which takes a pair of matrices as input and produces a block matrix * Matrix multiplication, Standard mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Inverse Theorem
In mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max A. Woodbury, says that the inverse of a rank-''k'' correction of some matrix can be computed by doing a rank-''k'' correction to the inverse of the original matrix. Alternative names for this formula are the matrix inversion lemma, Sherman–Morrison–Woodbury formula or just Woodbury formula. However, the identity appeared in several papers before the Woodbury report. The Woodbury matrix identity is : \left(A + UCV \right)^ = A^ - A^U \left(C^ + VA^U \right)^ VA^, where ''A'', ''U'', ''C'' and ''V'' are conformable matrices: ''A'' is ''n''×''n'', ''C'' is ''k''×''k'', ''U'' is ''n''×''k'', and ''V'' is ''k''×''n''. This can be derived using blockwise matrix inversion. While the identity is primarily used on matrices, it holds in a general ring or in an Ab-category. Discussion To prove this result, we will start by proving a simpler one. Replacing ''A'' and ''C'' with the ide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Woodbury Matrix Identity
In mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max A. Woodbury, says that the inverse of a rank-''k'' correction of some matrix can be computed by doing a rank-''k'' correction to the inverse of the original matrix. Alternative names for this formula are the matrix inversion lemma, Sherman–Morrison–Woodbury formula or just Woodbury formula. However, the identity appeared in several papers before the Woodbury report. The Woodbury matrix identity is : \left(A + UCV \right)^ = A^ - A^U \left(C^ + VA^U \right)^ VA^, where ''A'', ''U'', ''C'' and ''V'' are conformable matrices: ''A'' is ''n''×''n'', ''C'' is ''k''×''k'', ''U'' is ''n''×''k'', and ''V'' is ''k''×''n''. This can be derived using blockwise matrix inversion. While the identity is primarily used on matrices, it holds in a general ring or in an Ab-category. Discussion To prove this result, we will start by proving a simpler one. Replacing ''A'' and ''C'' with the ide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weinstein–Aronszajn Identity
In mathematics, the Weinstein–Aronszajn identity states that if A and B are matrices of size and respectively (either or both of which may be infinite) then, provided AB (and hence, also BA) is of trace class, :\det(I_m + AB) = \det(I_n + BA), where I_k is the identity matrix. It is closely related to the matrix determinant lemma and its generalization. It is the determinant analogue of the Woodbury matrix identity for matrix inverses. Proof The identity may be proved as follows. Let M be a matrix comprising the four blocks I_m, -A, B and I_n. :M = \begin I_m & -A \\ B & I_n \end. Because is invertible, the formula for the determinant of a block matrix gives :\det\begin I_m & -A \\ B & I_n \end = \det(I_m) \det\left(I_n - B I_m^ (-A)\right) = \det(I_n + BA). Because is invertible, the formula for the determinant of a block matrix gives :\det\begin I_m & -A\\ B & I_n \end = \det(I_n) \det\left(I_m - (-A) I_n^ B\right) = \det(I_m + AB). Thus :\det(I_n + B A) = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sherman–Morrison Formula
In mathematics, in particular linear algebra, the Sherman–Morrison formula, named after Jack Sherman and Winifred J. Morrison, computes the inverse of the sum of an invertible matrix A and the outer product, u v^\textsf, of vectors u and v. The Sherman–Morrison formula is a special case of the Woodbury formula. Though named after Sherman and Morrison, it appeared already in earlier publications. Statement Suppose A\in\mathbb^ is an invertible square matrix and u,v\in\mathbb^n are column vectors. Then A + uv^\textsf is invertible iff 1 + v^\textsf A^u \neq 0. In this case, :\left(A + uv^\textsf\right)^ = A^ - . Here, uv^\textsf is the outer product of two vectors u and v. The general form shown here is the one published by Bartlett. Proof (\Leftarrow) To prove that the backward direction 1 + v^\textsfA^u \neq 0 \Rightarrow A + uv^\textsf is invertible with inverse given as above) is true, we verify the properties of the inverse. A matrix Y (in this case the right-ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Vectors
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vector'', commonly denoted as d, is used to describe a unit vector being used to represent spatial direction and relative direction. 2D spatial directions are numerically equivalent to points on the unit circle and spatial directions in 3D are equivalent to a point on the unit sphere. The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e., :\mathbf = \frac where , u, is the norm (or length) of u. The term ''normalized vector'' is sometimes used as a synonym for ''unit vector''. Unit vectors are often chosen to form the basis of a vector space, and every vector in the space may be written as a linear combination of unit vectors. Orthogonal coordinates Cartesian coordinates Unit vectors may be us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computationally Expensive
In computer science, the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to execute them. Usually, this involves determining a function that relates the size of an algorithm's input to the number of steps it takes (its time complexity) or the number of storage locations it uses (its space complexity). An algorithm is said to be efficient when this function's values are small, or grow slowly compared to a growth in the size of the input. Different inputs of the same size may cause the algorithm to have different behavior, so best, worst and average case descriptions might all be of practical interest. When not otherwise specified, the function describing the performance of an algorithm is usually an upper bound, determined from the worst case inputs to the algorithm. The term "analysis of algorithms" was coined by Donald Knuth. Algorithm analysis is an important part of a broader ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjugate Matrix
In linear algebra, the adjugate or classical adjoint of a square matrix is the transpose of its cofactor matrix and is denoted by . It is also occasionally known as adjunct matrix, or "adjoint", though the latter today normally refers to a different concept, the adjoint operator which is the conjugate transpose of the matrix. The product of a matrix with its adjugate gives a diagonal matrix (entries not on the main diagonal are zero) whose diagonal entries are the determinant of the original matrix: :\mathbf \operatorname(\mathbf) = \det(\mathbf) \mathbf, where is the identity matrix of the same size as . Consequently, the multiplicative inverse of an invertible matrix can be found by dividing its adjugate by its determinant. Definition The adjugate of is the transpose of the cofactor matrix of , :\operatorname(\mathbf) = \mathbf^\mathsf. In more detail, suppose is a unital commutative ring and is an matrix with entries from . The -''minor'' of , denoted , is the determ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector (geometric)
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a '' directed line segment'', or graphically as an arrow connecting an ''initial point'' ''A'' with a ''terminal point'' ''B'', and denoted by \overrightarrow . A vector is what is needed to "carry" the point ''A'' to the point ''B''; the Latin word ''vector'' means "carrier". It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from ''A'' to ''B''. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often used to represent simple linear transformations, such as shearing or rotation. For example, if R is a square matrix representing a rotation (rotation matrix) and \mathbf is a column vector describing the position of a point in space, the product R\mathbf yields another column vector describing the position of that point after that rotation. If \mathbf is a row vector, the same transformation can be obtained using where R^ is the transpose of Main diagonal The entries a_ (''i'' = 1, …, ''n'') form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix. For instance, the main diagonal of the 4×4 matrix above contains the elements , , , . The d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
