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Determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix is denoted , , or . The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e & f \\ g & h & i \end= aei + bfg + cdh - ceg - bdi - afh. The determinant of a matrix can be defined in several equivalent ways. Leibniz formula expresses the determinant as a sum of signed products of matrix entries such that each summand is the product of different entries, and the number of these summands is n!, the factorial of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cramer's Rule
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-sides of the equations. It is named after Gabriel Cramer (1704–1752), who published the rule for an arbitrary number of unknowns in 1750, although Colin Maclaurin also published special cases of the rule in 1748 (and possibly knew of it as early as 1729). Cramer's rule implemented in a naive way is computationally inefficient for systems of more than two or three equations. In the case of equations in unknowns, it requires computation of determinants, while Gaussian elimination produces the result with the same computational complexity as the computation of a single determinant. Cramer's rule can also be n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobian Determinant
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature. Suppose is a function such that each of its first-order partial derivatives exist on . This function takes a point as input and produces the vector as output. Then the Jacobian matrix of is defined to be an matrix, denoted by , whose th entry is \mathbf J_ = \frac, or explicitly :\mathbf J = \begin \dfrac & \cdots & \dfrac \end = \begin \nabla^ f_1 \\ \vdots \\ \nabla^ f_m \end = \begin \dfrac & \cdots & \dfrac\\ \vdots & \ddots & \vdots\\ \dfrac & \cdo ... [...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 , , , . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Expansion
In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an matrix as a weighted sum of minors, which are the determinants of some submatrices of . Specifically, for every , \begin \det(B)&= \sum_^ (-1)^ B_ M_, \end where B_ is the entry of the th row and th column of , and M_ is the determinant of the submatrix obtained by removing the th row and the th column of . The term (-1)^ M_ is called the cofactor of B_ in . The Laplace expansion is often useful in proofs, as in, for example, allowing recursion on the size of matrices. It is also of didactic interest for its simplicity, and as one of several ways to view and compute the determinant. For large matrices, it quickly becomes inefficient to compute, when compared to Gaussian elimination. Examples Consider the matrix : B = \begin 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end. The determinant of this matrix can be computed by using ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invertible Matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix that satisfies the prior equation for a given invertible matrix . A square matrix that is ''not'' invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any finite region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices (-by- matrices for which ) do not ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Elimination
In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used to compute the rank of a matrix, the determinant of a square matrix, and the inverse of an invertible matrix. The method is named after Carl Friedrich Gauss (1777–1855) although some special cases of the method—albeit presented without proof—were known to Chinese mathematicians as early as circa 179 AD. To perform row reduction on a matrix, one uses a sequence of elementary row operations to modify the matrix until the lower left-hand corner of the matrix is filled with zeros, as much as possible. There are three types of elementary row operations: * Swapping two rows, * Multiplying a row by a nonzero number, * Adding a multiple of one row to another row. (subtraction can be achieved by multiplying one row with -1 and ad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Differential Form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression is an example of a -form, and can be integrated over an interval contained in the domain of : :\int_a^b f(x)\,dx. Similarly, the expression is a -form that can be integrated over a surface : :\int_S (f(x,y,z)\,dx\wedge dy + g(x,y,z)\,dz\wedge dx + h(x,y,z)\,dy\wedge dz). The symbol denotes the exterior product, sometimes called the ''wedge product'', of two differential forms. Likewise, a -form represents a volume element that can be integrated over a region of space. In general, a -form is an object that may be integrated over a -dimensional manifold, and is homogeneous of degree in the coordinate differentials dx, dy, \ldots. On an -dimensional manifold ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term '' rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, the four concepts—''parallelepiped'' and ''cube'' in three dimensions, ''parallelogram'' and ''square'' in two dimensions—are defined, but in the context of a more general affine geometry, in which angles are not differentiated, only ''parallelograms'' and ''parallelepipeds'' exist. Three equivalent definitions of ''parallelepiped'' are *a polyhedron with six faces ( hexahedron), each of which is a parallelogram, *a hexahedron with three pairs of parallel faces, and *a prism of which the base is a parallelogram. The rectangular cuboid (six rectangular faces), cube (six square faces), and the rhombohedron (six rhombus faces) are all specific cases of parallelepiped. "Parallelepiped" is now usually pronounced ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any base (that is, the characteristic polynomial does not depend on the choice of a basis). The characteristic equation, also known as the determinantal equation, is the equation obtained by equating the characteristic polynomial to zero. In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. Motivation In linear algebra, eigenvalues and eigenvectors play a fundamental role, since, given a linear transformation, an eigenvector is a vector whose direction is not changed by the transformation, and the correspondin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leibniz Formula For Determinants
In algebra, the Leibniz formula, named in honor of Gottfried Leibniz, expresses the determinant of a square matrix in terms of permutations of the matrix elements. If A is an n \times n matrix, where a_ is the entry in the i-th row and j-th column of A, the formula is :\det(A) = \sum_ \sgn(\tau) \prod_^n a_ = \sum_ \sgn(\sigma) \prod_^n a_ where \sgn is the sign function of permutations in the permutation group S_n, which returns +1 and -1 for even and odd permutations, respectively. Another common notation used for the formula is in terms of the Levi-Civita symbol and makes use of the Einstein summation notation, where it becomes : \det(A) = \epsilon_ _ \cdots _, which may be more familiar to physicists. Directly evaluating the Leibniz formula from the definition requires \Omega(n! \cdot n) operations in general—that is, a number of operations asymptotically proportional to n factorial—because n! is the number of order-n permutations. This is impractically difficult for eve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System Of Linear Equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in the three variables . A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by the ordered triple :(x,y,z)=(1,-2,-2), since it makes all three equations valid. The word "system" indicates that the equations are to be considered collectively, rather than individually. In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A sy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
