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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.
Augmented matrix. Matrix formed by appending columns of two other matrices. In linear algebra, an augmented matrix is a matrix obtained by appending a -dimensional column vector , on the right, as a further column to a -dimensional matrix . This is usually done for the purpose of performing the same elementary row operations on the augmented ...
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.
Rouché–Capelli theorem. Rouché–Capelli theorem is a theorem in linear algebra that determines the number of solutions for a system of linear equations, given the rank of its augmented matrix and coefficient matrix. The theorem is variously known as the: Frobenius theorem in the Czech Republic and in Slovakia.
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 ...
Transformation matrix. In linear algebra, linear transformations can be represented by matrices. If is a linear transformation mapping to and is a column vector with entries, then for some matrix , called the transformation matrix of . [citation needed] Note that has rows and columns, whereas the transformation is from to .
If the rows {v 1, ..., v k} are written as a matrix , then applying Gaussian elimination to the augmented matrix [|] will produce the orthogonalized vectors in place of . However the matrix A A T {\displaystyle AA^{\mathsf {T}}} must be brought to row echelon form , using only the row operation of adding a scalar multiple of one row to another ...
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 are the roots of the polynomial and the eigenvalues of . More broadly,any scalar-valued function is ...
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