adjointien

Adjointien is a term used in mathematics, particularly in the field of linear algebra, to describe a specific type of matrix or linear transformation. The adjoint of a square matrix A, denoted as adj(A), is defined as the transpose of the cofactor matrix of A. The cofactor matrix is obtained by replacing each element of A with its corresponding cofactor, which is the determinant of the minor matrix (the matrix obtained by removing the row and column of the element) multiplied by (-1)^(i+j), where i and j are the row and column indices of the element.

The adjoint matrix has several important properties. It is used in the computation of the inverse of a matrix. Specifically, if A is an invertible matrix, then its inverse can be expressed as (1/det(A)) * adj(A), where det(A) is the determinant of A. Additionally, the product of a matrix and its adjoint is a scalar matrix, specifically the determinant of the original matrix.

Adjointien is also applicable in the context of linear transformations. If T is a linear transformation represented by a matrix A, then the adjoint of T, denoted as T*, is the linear transformation represented by the adjoint matrix of A. The adjoint of a linear transformation has properties analogous to those of the adjoint of a matrix.

In summary, adjointien refers to the adjoint matrix or the adjoint linear transformation, which is a fundamental concept in linear algebra with applications in matrix inversion and the study of linear transformations.