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A matrix that is its own inverse (i.e., a matrix A such that A = A −1, and consequently A 2 = I), is called an involutory matrix. In relation to its adjugate. The adjugate of a matrix A can be used to find the inverse of A as follows: If A is an invertible matrix, then
The linear map h → J(x) ⋅ h is known as the derivative or the differential of f at x . When m = n, the Jacobian matrix is square, so its determinant is a well-defined function of x, known as the Jacobian determinant of f. It carries important information about the local behavior of f.
In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise 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 ...
Invariants of tensors. In mathematics, in the fields of multilinear algebra and representation theory, the principal invariants of the second rank tensor are the coefficients of the characteristic polynomial [1] where is the identity operator and represent the polynomial's eigenvalues . More broadly, any scalar-valued function is an invariant ...
Moore–Penrose inverse. In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix , often called the pseudoinverse, is the most widely known generalization of the inverse matrix. [1] It was independently described by E. H. Moore in 1920, [2] Arne Bjerhammar in 1951, [3] and Roger Penrose in 1955. [4]
In mathematics, the determinant is a scalar value that is a certain function of the entries of a square matrix. The determinant of a matrix A is commonly denoted det (A), det A, or |A|. Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero ...
Generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup. This article describes generalized inverses of a matrix . A matrix is a generalized inverse of a matrix if [1] [2] [3] A generalized inverse exists for an arbitrary matrix, and when a matrix has a regular inverse ...
Vandermonde matrix. In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row: an matrix. with entries , the jth power of the number , for all zero-based indices and . [1] Some authors define the Vandermonde matrix as the transpose of the above matrix.
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