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In linear algebra, the adjugate or classical adjoint of a square matrix A, adj (A), is the transpose of its cofactor matrix. [1][2] It is occasionally known as adjunct matrix, [3][4] or "adjoint", [5] though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose.
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix is an matrix obtained by transposing and applying complex conjugation to each entry (the complex conjugate of being , for real numbers and ). There are several notations, such as or , [1] , [2] or (often in physics) .
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 n × n - matrix B as a weighted sum of minors, which are the determinants of some (n − 1) × (n − 1) - submatrices of B. Specifically, for every i, the Laplace expansion ...
In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix generated from A by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors, which are useful for computing both the determinant and inverse of square matrices.
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 ...
Invertible matrix. In linear algebra, an invertible matrix is a square matrix which has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an inverse to undo the operation. Invertible matrices are the same size as their inverse.
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [1] If A is a differentiable map from the real numbers to n × n matrices, then. where tr (X) is the trace of the matrix X and is its adjugate matrix. (The latter equality only holds if A (t) is ...
A linear recursive sequence defined by for has the characteristic polynomial , whose transpose companion matrix generates the sequence: The vector is an eigenvector of this matrix, where the eigenvalue is a root of . Setting the initial values of the sequence equal to this vector produces a geometric sequence which satisfies the recurrence.