When black lesbians attempt to navigate pop culture's "gender-identity matrix", searching for their kindred's place in history, they often come up empty-handed.
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The identity matrix also has infinitely many non-symmetric square roots.
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The matrix associated to this form is the identity matrix. This is a Hermitian form.
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The factors are determined up to the negative 4th-order identity matrix, i.e. the central inversion.
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The general unitary group (also called the group of unitary similitudes) consists of all matrices A such that A∗A is a nonzero multiple of the identity matrix, and is just the product of the unitary group with the group of all positive multiples of the identity matrix.
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In mathematics, and especially gauge theory, Donaldson's theorem states that a definite intersection form of a compact, oriented, simply connected, smooth manifold of dimension 4 is diagonalisable. If the intersection form is positive (negative) definite, it can be diagonalized to the identity matrix (negative identity matrix) over the .
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Equation () can be stated equivalently as where is the by identity matrix and 0 is the zero vector.
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When A is m×n, it is a property of matrix multiplication that : I_m A = A I_n = A. In particular, the identity matrix serves as the unit of the ring of all n×n matrices, and as the identity element of the general linear group GL(n) (a group consisting of all invertible n×n matrices). In particular, the identity matrix is invertible—with its inverse being precisely itself. Where n×n matrices are used to represent linear transformations from an n-dimensional vector space to itself, In represents the identity function, regardless of the basis. The ith column of an identity matrix is the unit vector ei (the vector whose ith entry is 1 and 0 elsewhere) It follows that the determinant of the identity matrix is 1, and the trace is n.
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In general, a linear program will not be given in canonical form and an equivalent canonical tableau must be found before the simplex algorithm can start. This can be accomplished by the introduction of artificial variables. Columns of the identity matrix are added as column vectors for these variables. If the b value for a constraint equation is negative, the equation is negated before adding the identity matrix columns.
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To do this, the Cholesky decomposition is used to express Σ = A A'. Then the transformed vector Yi = A−1Xi has the identity matrix as its covariance matrix.
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That is, it is the only matrix such that: # When multiplied by itself, the result is itself # All of its rows and columns are linearly independent. The principal square root of an identity matrix is itself, and this is its only positive-definite square root. However, every identity matrix with at least two rows and columns has an infinitude of symmetric square roots.Mitchell, Douglas W. "Using Pythagorean triples to generate square roots of I2".
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A coherent algebra is an algebra of complex square matrices that is closed under ordinary matrix multiplication, Schur product, transposition, and contains both the identity matrix I and the all-ones matrix J.
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The only non-singular idempotent matrix is the identity matrix; that is, if a non-identity matrix is idempotent, its number of independent rows (and columns) is less than its number of rows (and columns). This can be seen from writing A^2 = A, assuming that has full rank (is non-singular), and pre- multiplying by A^{-1} to obtain A = IA = A^{-1}A^2 = A^{-1}A = I. When an idempotent matrix is subtracted from the identity matrix, the result is also idempotent. This holds since :(I-A)(I-A) = I-A-A+A^2 = I-A-A+A = I-A. A matrix is idempotent if and only if for all positive integers n, A^n = A. The 'if' direction trivially follows by taking n=2.
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A symmetric diagonal matrix can be defined as a matrix that is both upper- and lower-triangular. The identity matrix In and any square zero matrix are diagonal. A one-dimensional matrix is always diagonal.
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This is equivalent to saying that A times its transpose must be the identity matrix. If these conditions do not hold, the formula describes a more general affine transformation of the plane provided that the determinant of A is not zero. The formula defines a translation if and only if A is the identity matrix. The transformation is a rotation around some point if and only if A is a rotation matrix, meaning that : A_{1 1} A_{2 2} - A_{2 1} A_{1 2} = 1 .
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In the fields of machine learning, the theory of computation, and random matrix theory, a probability distribution over vectors is said to be in isotropic position if its covariance matrix is equal to the identity matrix.
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The projective linear group and the projective special linear group are the quotients of and by their centers (which consist of the multiples of the identity matrix therein); they are the induced action on the associated projective space.
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Reflection through the origin is an orthogonal transformation corresponding to scalar multiplication by -1, and can also be written as -I, where I is the identity matrix. In three dimensions, this sends (x, y, z) \mapsto (-x, -y, -z), and so forth.
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This significantly speeds up the often real time calculations of the filter. In the case when C is the identity matrix I, the matrix I+VA^{-1}U is known in numerical linear algebra and numerical partial differential equations as the capacitance matrix.
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The Identity Matrix is a science fiction novel by American writer Jack L. Chalker, published in 1982 by Timescape Books. The work focuses on the body swap and enemy mine plot devices, as well as a background conflict between two powerful alien races.
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In mathematics, an integer matrix is a matrix whose entries are all integers. Examples include binary matrices, the zero matrix, the matrix of ones, the identity matrix, and the adjacency matrices used in graph theory, amongst many others. Integer matrices find frequent application in combinatorics.
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Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which the errors covariance matrix is allowed to be different from an identity matrix. WLS is also a specialization of generalized least squares in which the above matrix is diagonal.
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In numerical analysis, interpolative decomposition (ID) factors a matrix as the product of two matrices, one of which contains selected columns from the original matrix, and the other of which has a subset of columns consisting of the identity matrix and all its values are no greater than 2 in absolute value.
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If M is an idempotent matrix, meaning that MM = M, then if it is not the identity matrix, its determinant is zero, and its trace equals its rank, which (excluding the zero matrix) is 1. Then the above formula has s = 0 and τ = 1, giving M and −M as two square roots of M.
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A square matrix A is called invertible or non-singular if there exists a matrix B such that :AB = BA = I , where I is the n×n identity matrix with 1s on the main diagonal and 0s elsewhere. If B exists, it is unique and is called the inverse matrix of A, denoted A.
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For orthonormal cartesian coordinates, the covariant and contravariant basis are identical, since the basis set in this case is just the identity matrix, however, for non-affine coordinate system such as polar or spherical is a need to distinguish between decomposition by use of contravariant or covariant basis set for generating the components of the coordinate system.
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This implies that the submatrix of the m + n − 2i first rows of the column echelon form of Ti is the identity matrix and thus that si is not 0. Thus Si is a polynomial in the image of \varphi_i, which is a multiple of the GCD and has the same degree. It is thus a greatest common divisor.
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Let , the n dimensional real vector space. Then the standard dot product is a symmetric bilinear form, . The matrix corresponding to this bilinear form (see below) on a standard basis is the identity matrix. Let V be any vector space (including possibly infinite- dimensional), and assume T is a linear function from V to the field.
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In mathematics, an involutory matrix is a matrix that is its own inverse. That is, multiplication by matrix A is an involution if and only if A2 = I. Involutory matrices are all square roots of the identity matrix. This is simply a consequence of the fact that any nonsingular matrix multiplied by its inverse is the identity..
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The proof is bijective: a matrix is an adjacency matrix of a DAG if and only if is a (0,1) matrix with all eigenvalues positive, where denotes the identity matrix. Because a DAG cannot have self-loops, its adjacency matrix must have a zero diagonal, so adding preserves the property that all matrix coefficients are 0 or 1., Article 04.3.3.
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In a noncommutative gauge theory, the ADHM construction is identical but the moment map \vec\mu is set equal to the self-dual projection of the noncommutativity matrix of the spacetime times the identity matrix. In this case instantons exist even when the gauge group is U(1). The noncommutative instantons were discovered by Nikita Nekrasov and Albert Schwarz in 1998.
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Let I denote the identity matrix and let J denote the matrix of ones, both matrices of order v. The adjacency matrix A of a strongly regular graph satisfies two equations. First: :AJ = JA = kJ, which is a trivial restatement of the regularity requirement. This shows that k is an eigenvalue of the adjacency matrix with the all-ones eigenvector.
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The set of all invertible diagonal matrices forms a subgroup of isomorphic to (F×)n. In fields like R and C, these correspond to rescaling the space; the so-called dilations and contractions. A scalar matrix is a diagonal matrix which is a constant times the identity matrix. The set of all nonzero scalar matrices forms a subgroup of isomorphic to F× .
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For large samples, the shrinkage intensity will reduce to zero, hence in this case the shrinkage estimator will be identical to the empirical estimator. Apart from increased efficiency the shrinkage estimate has the additional advantage that it is always positive definite and well conditioned. Various shrinkage targets have been proposed: # the identity matrix, scaled by the average sample variance; # the single-index model; # the constant-correlation model, where the sample variances are preserved, but all pairwise correlation coefficients are assumed to be equal to one another; # the two-parameter matrix, where all variances are identical, and all covariances are identical to one another (although not identical to the variances); # the diagonal matrix containing sample variances on the diagonal and zeros everywhere else; # the identity matrix. The shrinkage estimator can be generalized to a multi-target shrinkage estimator that utilizes several targets simultaneously.
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The general linear group GL(2, 7) consists of all invertible 2×2 matrices over F7, the finite field with 7 elements. These have nonzero determinant. The subgroup SL(2, 7) consists of all such matrices with unit determinant. Then PSL(2, 7) is defined to be the quotient group :SL(2, 7)/{I, −I} obtained by identifying I and −I, where I is the identity matrix.
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Left- and right-isoclinic rotations are represented respectively by left- and right- multiplication by unit quaternions; see the paragraph "Relation to quaternions" below. The four rotations are pairwise different except if or . The angle corresponds to the identity rotation; corresponds to the central inversion, given by the negative of the identity matrix. These two elements of SO(4) are the only ones that are simultaneously left- and right-isoclinic.
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The correct formula is: HH^T=I_n, where In is the n×n identity matrix and HT is the transpose of H. In the 1999 paper, the authors generalize the Reverse Jacket matrix [RJ]N using Hadamard matrices and Weighted Hadamard matrices.Lee, Seung-Rae, and Moon Ho Lee. "On the Reverse Jacket matrix for weighted Hadamard transform." IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, Vol.
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Extended BASIC added the suite of matrix math operations from Dartmouth BASIC's Fifth Edition. These were, in essence, macros that performed operations that would otherwise be accomplished with loops. The system included a number of pre-rolled matrixes, like for a zero- matrix, for a matrix of all 1's, for the identity matrix. Most mathematical operations were supported, for instance, multiplies every element in A by 2.
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An elliptical distribution with a zero mean and variance in the form \alpha I where I is the identity-matrix is called a spherical distribution. For spherical distributions, classical results on parameter-estimation and hypothesis-testing hold have been extended. Similar results hold for linear models, and indeed also for complicated models ( especially for the growth curve model). The analysis of multivariate models uses multilinear algebra (particularly Kronecker products and vectorization) and matrix calculus.
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Specifically, if we choose an orthonormal basis of \R^3, every rotation is described by an orthogonal 3×3 matrix (i.e. a 3×3 matrix with real entries which, when multiplied by its transpose, results in the identity matrix) with determinant 1. The group SO(3) can therefore be identified with the group of these matrices under matrix multiplication. These matrices are known as "special orthogonal matrices", explaining the notation SO(3).
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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 is of trace class (and hence, so is BA), :\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.
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In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GLn(R) when R is a field. Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations. Elementary row operations are used in Gaussian elimination to reduce a matrix to row echelon form.
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Every logical matrix a = ( a i j ) has an transpose aT = ( a j i ). Suppose a is a logical matrix with no columns or rows identically zero. Then the matrix product, using Boolean arithmetic, aT a contains the m × m identity matrix, and the product a aT contains the n × n identity. As a mathematical structure, the Boolean algebra U forms a lattice ordered by inclusion; additionally it is a multiplicative lattice due to matrix multiplication.
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In mathematics, a weighing matrix W of order n and weight w is an n × n (0,1,-1)-matrix such that WW^{T}=wI_n, where W^T is the transpose of W and I_n is the identity matrix of order n. For convenience, a weighing matrix of order n and weight w is often denoted by W(n,w). A W(n,n) is a Hadamard matrix and a W(n,n-1) is equivalent to a conference matrix.
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Using the exact inverse of A would be nice but finding the inverse of a matrix is something we want to avoid because of the computational expense. Now, since PA ≈ I where I is the identity matrix, the eigenvalues of PA should all be close to 1. By the Gershgorin circle theorem, every eigenvalue of PA lies within a known area and so we can form a rough estimate of how good our choice of P was.
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An M-matrix is commonly defined as follows: Definition: Let be a real Z-matrix. That is, where for all . Then matrix A is also an M-matrix if it can be expressed in the form , where with , for all , where is at least as large as the maximum of the moduli of the eigenvalues of , and is an identity matrix. For the non-singularity of , according to the Perron-Frobenius theorem, it must be the case that .
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Moreover, since A and B are Hermitian matrices, their eigenvalues are all real numbers. If λ1(B) is the maximum eigenvalue of B and λn(A) the minimum eigenvalue of A, a sufficient criterion to have A ≥ B is that λn(A) ≥ λ1(B). If A or B is a multiple of the identity matrix, then this criterion is also necessary. The Loewner order does not have the least-upper-bound property, and therefore does not form a lattice.
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The term is also sometimes used informally to mean a vector, matrix, tensor, or other usually "compound" value that is actually reduced to a single component. Thus, for example, the product of a 1×n matrix and an n×1 matrix, which is formally a 1×1 matrix, is often said to be a scalar. The term scalar matrix is used to denote a matrix of the form kI where k is a scalar and I is the identity matrix.
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The variable for this column is now a basic variable, replacing the variable which corresponded to the r-th column of the identity matrix before the operation. In effect, the variable corresponding to the pivot column enters the set of basic variables and is called the entering variable, and the variable being replaced leaves the set of basic variables and is called the leaving variable. The tableau is still in canonical form but with the set of basic variables changed by one element.
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His work included producing, directing, writing, editing, cinematography and sound recording. In 1983, he turned to drama, producing and directing the premiere of HBO's Family Playhouse and a special for American Playhouse. That year, he co-created and produced the family action-adventure television series Danger Bay; the hit CBC–Disney Channel series ran for six years and 123 episodes. Since then he has produced television series such as My Secret Identity, Matrix and Max Glick, as well as miniseries and movies of the week.
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This maps virtual position in a document to istream positions in the pooled content that the document is built from. The POOM starts out an identity matrix, then each edit to the document slices and rearranges horizontal strips of the map. The POOM can be queried in the V->I or I->V directions by searching in squat, wide address ranges or tall, narrow ones. The Spanfilade collects the union of all spans of istream content used by a document or set of documents.
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If is a linear endomorphism of a vector space over a field , an eigenvector of is a nonzero vector of such that for some scalar in . This scalar is an eigenvalue of . If the dimension of is finite, and a basis has been chosen, and may be represented, respectively, by a square matrix and a column matrix ; the equation defining eigenvectors and eigenvalues becomes :Mz=az. Using the identity matrix , whose entries are all zero, except those of the main diagonal, which are equal to one, this may be rewritten :(M-aI)z=0.
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When both and are matrices, the trace of the (ring-theoretic) commutator of and vanishes: , because and is linear. One can state this as "the trace is a map of Lie algebras from operators to scalars", as the commutator of scalars is trivial (it is an Abelian Lie algebra). In particular, using similarity invariance, it follows that the identity matrix is never similar to the commutator of any pair of matrices. Conversely, any square matrix with zero trace is a linear combinations of the commutators of pairs of matrices.
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In mathematics, a conference matrix (also called a C-matrix) is a square matrix C with 0 on the diagonal and +1 and −1 off the diagonal, such that CTC is a multiple of the identity matrix I. Thus, if the matrix has order n, CTC = (n−1)I. Some authors use a more general definition, which requires there to be a single 0 in each row and column but not necessarily on the diagonal. Conference matrices first arose in connection with a problem in telephony.Belevitch, pp. 231-244.
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One is to use a pseudo inverse instead of the usual matrix inverse in the above formulae. However, better numeric stability may be achieved by first projecting the problem onto the subspace spanned by \Sigma_b . Another strategy to deal with small sample size is to use a shrinkage estimator of the covariance matrix, which can be expressed mathematically as : \Sigma = (1-\lambda) \Sigma+\lambda I\, where I is the identity matrix, and \lambda is the shrinkage intensity or regularisation parameter. This leads to the framework of regularized discriminant analysis or shrinkage discriminant analysis.
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The geometrical operation of moving from a basic feasible solution to an adjacent basic feasible solution is implemented as a pivot operation. First, a nonzero pivot element is selected in a nonbasic column. The row containing this element is multiplied by its reciprocal to change this element to 1, and then multiples of the row are added to the other rows to change the other entries in the column to 0. The result is that, if the pivot element is in row r, then the column becomes the r-th column of the identity matrix.
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The interpretation of the matrices is that they act as generators of motions on the space of states. For example, the motion generated by P can be found by solving the Heisenberg equation of motion using P as a Hamiltonian, : dX = i[X,P] ds = ds \, : dP = i[P,P] ds = 0 \, . These are translations of the matrix X by a multiple of the identity matrix, :X\rightarrow X+s I ~. This is the interpretation of the derivative operator D: , the exponential of a derivative operator is a translation (so Lagrange's shift operator).
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In mathematics, the projective unitary group is the quotient of the unitary group by the right multiplication of its center, , embedded as scalars. Abstractly, it is the holomorphic isometry group of complex projective space, just as the projective orthogonal group is the isometry group of real projective space. In terms of matrices, elements of are complex unitary matrices, and elements of the center are diagonal matrices equal to multiplied by the identity matrix. Thus, elements of correspond to equivalence classes of unitary matrices under multiplication by a constant phase .
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We consider an n×n matrix A. The characteristic polynomial of A, denoted by pA(t), is the polynomial defined by :p_A(t) = \det \left(tI - A\right) where I denotes the n×n identity matrix. Some authors define the characteristic polynomial to be . That polynomial differs from the one defined here by a sign , so it makes no difference for properties like having as roots the eigenvalues of A; however the definition above always gives a monic polynomial, whereas the alternative definition is monic only when n is even.
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Let D be the set of diagonal matrices in the matrix ring Mn(R), that is the set of the matrices such that every nonzero entry, if any, is on the main diagonal. Then D is closed under matrix addition and matrix multiplication, and contains the identity matrix, so it is a subalgebra of Mn(R). As an algebra over R, D is isomorphic to the direct product of n copies of R. It is a free R-module of dimension n. The idempotent elements of D are the diagonal matrices such that the diagonal entries are themselves idempotent.
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In fact, R only needs to be a semiring for Mn(R) to be defined. In this case, Mn(R) is a semiring, called the matrix semiring'. Similarly, if R is a commutative semiring, then Mn(R) is a '. For example, if R is the Boolean semiring (the two-element Boolean algebra R = {0,1} with 1 + 1 = 1), then Mn(R) is the semiring of binary relations on an n-element set with union as addition, composition of relations as multiplication, the empty relation (zero matrix) as the zero, and the identity relation (identity matrix) as the unit.
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Since addition and multiplication of matrices have all needed properties for field operations except for commutativity of multiplication and existence of multiplicative inverses, one way to verify if a set of matrices is a field with the usual operations of matrix sum and multiplication is to check whether # the set is closed under addition, subtraction and multiplication; # the neutral element for matrix addition (that is, the zero matrix) is included; # multiplication is commutative; # the set contains a multiplicative identity (note that this does not have to be the identity matrix); and # each matrix that is not the zero matrix has a multiplicative inverse.
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For example, `Fraction` is a function, that takes an `IntegralDomain` as argument, and returns the field of fractions of its argument. As another example, the ring of 4\times 4 matrices with rational entries would be constructed as `SquareMatrix(4, Fraction Integer)`. Of course, when working in this domain, `1` is interpreted as the identity matrix and `A^-1` would give the inverse of the matrix `A`, if it exists. Several operations can have the same name, and the types of both the arguments and the result are used to determine which operation is applied (cf.
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The matrix representation of the equality relation on a finite set is the identity matrix I, that is, the matrix whose entries on the diagonal are all 1, while the others are all 0. More generally, if relation R satisfies I ⊂ R, then R is a reflexive relation. If the Boolean domain is viewed as a semiring, where addition corresponds to logical OR and multiplication to logical AND, the matrix representation of the composition of two relations is equal to the matrix product of the matrix representations of these relations. This product can be computed in expected time O(n2).
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The powers of , obtained by substitution from powers of , are defined by repeated matrix multiplication; the constant term of gives a multiple of the power 0, which is defined as the identity matrix. The theorem allows to be expressed as a linear combination of the lower matrix powers of . When the ring is a field, the Cayley–Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial. The theorem was first proved in 1853 in terms of inverses of linear functions of quaternions, a non-commutative ring, by Hamilton.
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In some applications, an orthogonalization method such as the Gram–Schmidt process is performed in order to produce a set of orthogonal basis functions. This can in principle save computational time when the computer is solving the Roothaan–Hall equations by converting the overlap matrix effectively to an identity matrix. However, in most modern computer programs for molecular Hartree–Fock calculations this procedure is not followed due to the high numerical cost of orthogonalization and the advent of more efficient, often sparse, algorithms for solving the generalized eigenvalue problem, of which the Roothaan–Hall equations are an example.
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A three-dimensional rotation, with an axis of rotation along the -axis and a plane of rotation in the -plane In three- dimensional space there are an infinite number of planes of rotation, only one of which is involved in any given rotation. That is, for a general rotation there is precisely one plane which is associated with it or which the rotation takes place in. The only exception is the trivial rotation, corresponding to the identity matrix, in which no rotation takes place. In any rotation in three dimensions there is always a fixed axis, the axis of rotation.
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In other forms of digital tomography, even less information about each row or column is given: only the total number of squares, rather than the number and length of the blocks of squares. An equivalent version of the problem is that we must recover a given 0-1 matrix given only the sums of the values in each row and in each column of the matrix. Although there exist polynomial time algorithms to find a matrix having given row and column sums,. the solution may be far from unique: any submatrix in the form of a 2 × 2 identity matrix can be complemented without affecting the correctness of the solution.
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One can, for example, modify the Hessian by adding a correction matrix B_k so as to make f(x_k) + B_k positive definite. One approach is to diagonalize the Hessian and choose B_k so that f(x_k) + B_k has the same eigenvectors as the Hessian, but with each negative eigenvalue replaced by \epsilon>0. An approach exploited in the Levenberg–Marquardt algorithm (which uses an approximate Hessian) is to add a scaled identity matrix to the Hessian, \mu I, with the scale adjusted at every iteration as needed. For large \mu and small Hessian, the iterations will behave like gradient descent with step size 1/\mu.
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A whitening transformation or sphering transformation is a linear transformation that transforms a vector of random variables with a known covariance matrix into a set of new variables whose covariance is the identity matrix, meaning that they are uncorrelated and each have variance 1. The transformation is called "whitening" because it changes the input vector into a white noise vector. Several other transformations are closely related to whitening: # the decorrelation transform removes only the correlations but leaves variances intact, # the standardization transform sets variances to 1 but leaves correlations intact, # a coloring transformation transforms a vector of white random variables into a random vector with a specified covariance matrix.
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Given an oriented "contour" Σ (technically: an oriented union of smooth curves without points of infinite self-intersection in the complex plane). A Birkhoff factorization problem is the following. Given a matrix function V defined on the contour Σ, to find a holomorphic matrix function M defined on the complement of Σ, such that two conditions be satisfied: # If M+ and M− denote the non-tangential limits of M as we approach Σ, then M+ = M−V, at all points of non-intersection in Σ. #As z tends to infinity along any direction outside Σ, M tends to the identity matrix. In the simplest case V is smooth and integrable.
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In algebra, an Okubo algebra or pseudo-octonion algebra is an 8-dimensional non-associative algebra similar to the one studied by Susumu Okubo. Okubo algebras are composition algebras, flexible algebras (A(BA) = (AB)A), Lie admissible algebras, and power associative, but are not associative, not alternative algebras, and do not have an identity element. Okubo's example was the algebra of 3-by-3 trace-zero complex matrices, with the product of X and Y given by aXY + bYX – Tr(XY)I/3 where I is the identity matrix and a and b satisfy a + b = 3ab = 1\. The Hermitian elements form an 8-dimensional real non-associative division algebra.
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The infinite general linear group or stable general linear group is the direct limit of the inclusions as the upper left block matrix. It is denoted by either GL(F) or , and can also be interpreted as invertible infinite matrices which differ from the identity matrix in only finitely many places. It is used in algebraic K-theory to define K1, and over the reals has a well-understood topology, thanks to Bott periodicity. It should not be confused with the space of (bounded) invertible operators on a Hilbert space, which is a larger group, and topologically much simpler, namely contractible - see Kuiper's theorem.
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It can be proved that two matrices are similar if and only if one can transform one in the other by elementary row and column operations. For a matrix representing a linear map from to , the row operations correspond to change of bases in and the column operations correspond to change of bases in . Every matrix is similar to an identity matrix possibly bordered by zero rows and zero columns. In terms of vector spaces, this means that, for any linear map from to , there are bases such that a part of the basis of is mapped bijectively on a part of the basis of , and that the remaining basis elements of , if any, are mapped to zero.
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Ricci calculus, and index notation more generally, distinguishes between lower indices (subscripts) and upper indices (superscripts); the latter are not exponents, even though they may look as such to the reader only familiar with other parts of mathematics. It is in special cases (that the metric tensor is everywhere equal to the identity matrix) possible to drop the distinction between upper and lower indices, and then all indices could be written in the lower position – coordinate formulae in linear algebra such as a_{ij} b_{jk} for the product of matrices can sometimes be understood as examples of this – but in general the notation requires that the distinction between upper and lower indices is observed and maintained.
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In the trivial case, S = I, where I is the identity matrix, gives regular OFDM without spreading. The received signal can also be expressed as: r = F−1ΛHFF−1(ΛCF)b, where S = ΛCF, and C is a circulant matrix defined by C = F−1ΛCF, where ΛC is the circulant’s diagonal matrix. Thus, the received signal, r, can be written as r = F−1ΛHΛCFb = F−1ΛCΛHFb, and the signal y after the receiver's DFT is y = ΛCΛHFb The spreading matrix S can include a pre-equalization diagonal matrix (e.g., ΛC = ΛH−1 in the case of zero-forcing), or equalization can be performed at the receiver between the DFT (OFDM demodulator) and the inverse- DFT (CI de-spreader).
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The existence of a symmetric (v, b, r, k, λ)-design is equivalent to the existence of a v × v incidence matrix R with elements 0 and 1 satisfying : where is the v × v identity matrix and J is the v × v all-1 matrix. In essence, the Bruck–Ryser–Chowla theorem is a statement of the necessary conditions for the existence of a rational v × v matrix R satisfying this equation. In fact, the conditions stated in the Bruck–Ryser–Chowla theorem are not merely necessary, but also sufficient for the existence of such a rational matrix R. They can be derived from the Hasse–Minkowski theorem on the rational equivalence of quadratic forms.
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As described in the previous sections, the Isolation Forest algorithm performs very well from both the computational and the memory consumption points of view. The main problem with the original algorithm is that the way the branching of trees takes place introduce a bias, which is likely to reduce the reliability of the anomaly scores for ranking the data. This is the main motivation behind the introduction of the Extended Isolation Forest (EIF) algorithm by Hariri et al.left In order to understand why the original Isolation Forest suffers from that bias, the authors provide a practical example based on a random dataset taken from a 2-D normal distribution with zero mean and covariance given by the identity matrix.
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This leads to the general result, : The hyperdeterminant of format (k_1,\ldots,k_r) is an invariant under an action of the group SL(k_1+1) \otimes \cdots \otimes SL(k_r+1) E.g. the determinant of an n × n matrix is an SL(n)2 invariant and Cayley's hyperdeterminant for a 2×2×2 hypermatrix is an SL(2)3 invariant. A more familiar property of a determinant is that if you add a multiple of a row (or column) to a different row (or column) of a square matrix then its determinant is unchanged. This is a special case of its invariance in the case where the special linear transformation matrix is an identity matrix plus a matrix with only one non- zero off-diagonal element.
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An Hadamard matrix of size m is an m × m matrix H whose entries are ±1 such that HH⊤ = mIm, where H⊤ is the transpose of H and Im is the m × m identity matrix. An Hadamard matrix can be put into standardized form (that is, converted to an equivalent Hadamard matrix) where the first row and first column entries are all +1. If the size m > 2 then m must be a multiple of 4. Given an Hadamard matrix of size 4a in standardized form, remove the first row and first column and convert every −1 to a 0. The resulting 0–1 matrix M is the incidence matrix of a symmetric 2-(4a − 1, 2a − 1, a − 1) design called an Hadamard 2-design.
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An almost complex structure on a real 2n-manifold is a GL(n, C)-structure (in the sense of G-structures) – that is, the tangent bundle is equipped with a linear complex structure. Concretely, this is an endomorphism of the tangent bundle whose square is −I; this endomorphism is analogous to multiplication by the imaginary number i, and is denoted J (to avoid confusion with the identity matrix I). An almost complex manifold is necessarily even-dimensional. An almost complex structure is weaker than a complex structure: any complex manifold has an almost complex structure, but not every almost complex structure comes from a complex structure. Note that every even-dimensional real manifold has an almost complex structure defined locally from the local coordinate chart.
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Lattice reduction algorithms are used in a number of modern number theoretical applications, including in the discovery of a spigot algorithm for \pi. Although determining the shortest basis is possibly an NP-complete problem, algorithms such as the LLL algorithm can find a short (not necessarily shortest) basis in polynomial time with guaranteed worst-case performance. LLL is widely used in the cryptanalysis of public key cryptosystems. When used to find integer relations, a typical input to the algorithm consists of an augmented n x n identity matrix with the entries in the last column consisting of the n elements (multiplied by a large positive constant w to penalize vectors that do not sum to zero) between which the relation is sought.
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Binary relations over sets X and Y can be represented algebraically by logical matrices indexed by X and Y with entries in the Boolean semiring (addition corresponds to OR and multiplication to AND) where matrix addition corresponds to union of relations, matrix multiplication corresponds to composition of relations (of a relation over X and Y and a relation over Y and Z), the Hadamard product corresponds to intersection of relations, the zero matrix corresponds to the empty relation, and the matrix of ones corresponds to the universal relation. Homogeneous relations (when ) form a matrix semiring (indeed, a matrix semialgebra over the Boolean semiring) where the identity matrix corresponds to the identity relation.Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series.
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In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with identity matrix as the identity element of the group. The group is so named because the columns of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position. To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix.
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The unique solution λ represents the rate of growth of the economy, which equals the interest rate. Proving the existence of a positive growth rate and proving that the growth rate equals the interest rate were remarkable achievements, even for von Neumann.For this problem to have a unique solution, it suffices that the nonnegative matrices A and B satisfy an irreducibility condition, generalizing that of the Perron–Frobenius theorem of nonnegative matrices, which considers the (simplified) eigenvalue problem : A - λ I q = 0, where the nonnegative matrix A must be square and where the diagonal matrix I is the identity matrix. Von Neumann's irreducibility condition was called the "whales and wranglers" hypothesis by David Champernowne, who provided a verbal and economic commentary on the English translation of von Neumann's article.
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The unique solution λ represents the growth factor which is 1 plus the rate of growth of the economy; the rate of growth equals the interest rate.For this problem to have a unique solution, it suffices that the nonnegative matrices A and B satisfy an irreducibility condition, generalizing that of the Perron–Frobenius theorem of nonnegative matrices, which considers the (simplified) eigenvalue problem : A − λ I q = 0, where the nonnegative matrix A must be square and where the diagonal matrix I is the identity matrix. Von Neumann's irreducibility condition was called the "whales and wranglers" hypothesis by D. G. Champernowne, who provided a verbal and economic commentary on the English translation of von Neumann's article. Von Neumann's hypothesis implied that every economic process used a positive amount of every economic good.
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The tableau form used above to describe the algorithm lends itself to an immediate implementation in which the tableau is maintained as a rectangular (m + 1)-by-(m + n + 1) array. It is straightforward to avoid storing the m explicit columns of the identity matrix that will occur within the tableau by virtue of B being a subset of the columns of [A, I]. This implementation is referred to as the "standard simplex algorithm". The storage and computation overhead are such that the standard simplex method is a prohibitively expensive approach to solving large linear programming problems. In each simplex iteration, the only data required are the first row of the tableau, the (pivotal) column of the tableau corresponding to the entering variable and the right-hand-side.
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A tight lower bound is not known on the number of required additions, although lower bounds have been proved under some restrictive assumptions on the algorithms. In 1973, Morgenstern proved an Ω(N log N) lower bound on the addition count for algorithms where the multiplicative constants have bounded magnitudes (which is true for most but not all FFT algorithms). This result, however, applies only to the unnormalized Fourier transform (which is a scaling of a unitary matrix by a factor of \sqrt N), and does not explain why the Fourier matrix is harder to compute than any other unitary matrix (including the identity matrix) under the same scaling. Pan (1986) proved an Ω(N log N) lower bound assuming a bound on a measure of the FFT algorithm's "asynchronicity", but the generality of this assumption is unclear.
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Homography groups also called projective linear groups are denoted when acting on a projective space of dimension n over a field F. Above definition of homographies shows that may be identified to the quotient group , where is the general linear group of the invertible matrices, and F×I is the group of the products by a nonzero element of F of the identity matrix of size . When F is a Galois field GF(q) then the homography group is written . For example, acts on the eight points in the projective line over the finite field GF(7), while , which is isomorphic to the alternating group A5, is the homography group of the projective line with five points. The homography group is a subgroup of the collineation group of the collineations of a projective space of dimension n.
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In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex field) satisfies its own characteristic equation. If is a given matrix and is the identity matrix, then the characteristic polynomial of is defined as ::p(\lambda)=\det(\lambda I_n-A)~, where is the determinant operation and is a variable for a scalar element of the base ring. Since the entries of the matrix (\lambda I_n-A) are (linear or constant) polynomials in , the determinant is also an -th order monic polynomial in . The Cayley–Hamilton theorem states that if one defines an analogous matrix equation, , consisting of the replacement of the scalar variable with the matrix , then this polynomial in the matrix results in the zero matrix, ::p(A)= 0.
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It is sufficient to show that given any k \times l matrix M, where k is greater than or equal to l, such that the rank of M is l, for all x\in F_2^k, xM takes every value in F_2^l the same number of times. Since M has rank l, we can write M as two matrices of the same size, M_1 and M_2, where M_1 has rank equal to l. This means that xM can be rewritten as x_1M_1 + x_2M_2 for some x_1 and x_2. If we consider M written with respect to a basis where the first l rows are the identity matrix, then x_1 has zeros wherever M_2 has nonzero rows, and x_2 has zeros wherever M_1 has nonzero rows. Now any value y, where y=xM, can be written as x_1M_1+x_2M_2 for some vectors x_1, x_2.
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The determinant of a matrix product of square matrices equals the product of their determinants: :\det(AB) = \det (A) \times \det (B) Thus the determinant is a multiplicative map. This property is a consequence of the characterization given above of the determinant as the unique n-linear alternating function of the columns with value 1 on the identity matrix, since the function that maps can easily be seen to be n-linear and alternating in the columns of M, and takes the value det(A) at the identity. The formula can be generalized to (square) products of rectangular matrices, giving the Cauchy–Binet formula, which also provides an independent proof of the multiplicative property. The determinant det(A) of a matrix A is non-zero if and only if A is invertible or, yet another equivalent statement, if its rank equals the size of the matrix.
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Like the characteristic polynomial, the minimal polynomial does not depend on the base field, in other words considering the matrix as one with coefficients in a larger field does not change the minimal polynomial. The reason is somewhat different from for the characteristic polynomial (where it is immediate from the definition of determinants), namely the fact that the minimal polynomial is determined by the relations of linear dependence between the powers of : extending the base field will not introduce any new such relations (nor of course will it remove existing ones). The minimal polynomial is often the same as the characteristic polynomial, but not always. For example, if is a multiple of the identity matrix, then its minimal polynomial is since the kernel of is already the entire space; on the other hand its characteristic polynomial is (the only eigenvalue is , and the degree of the characteristic polynomial is always equal to the dimension of the space).
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Then we have :α( ar − br ) = ar+1 − br, where α = exp( i θ ) for some θ (here i is the square root of −1). This yields the following expression to compute the br ' s: :br = (1−α)−1 ( ar+1 − αar ). In terms of the linear operator S : Cn → Cn that cyclically permutes the coordinates one place, we have :B = (1−α)−1( S − αI )A, where I is the identity matrix. This means that the polygon An−2 that we need to show is regular is obtained from A0 by applying the composition of the following operators: : ( 1 − ωk )−1( S − ωk I ) for k = 1, 2, ... , n − 2, where ω = exp( 2πi/n ). (These commute because they are all polynomials in the same operator S.) A polygon P = ( p1, p2, ..., pn ) is a regular n-gon if each side of P is obtained from the next by rotating through an angle of 2π/n, that is, if : pr + 1 − pr = ω( pr + 2 − pr + 1 ).
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