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  2. Laplace expansion - Wikipedia

    en.wikipedia.org/wiki/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 along the ith row is ...

  3. Minor (linear algebra) - Wikipedia

    en.wikipedia.org/wiki/Minor_(linear_algebra)

    In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down 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 in turn are useful for computing both the determinant and inverse of square matrices.

  4. Adjugate matrix - Wikipedia

    en.wikipedia.org/wiki/Adjugate_matrix

    Adjugate matrix. In linear algebra, the adjugate of a square matrix A is the transpose of its cofactor matrix and is denoted by adj (A). [1][2] It is also occasionally known as adjunct matrix, [3][4] or "adjoint", [5] though the latter term today normally refers to a different concept, the adjoint operator which for a matrix is the conjugate ...

  5. Jacobi's formula - Wikipedia

    en.wikipedia.org/wiki/Jacobi's_formula

    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 ...

  6. Cramer's rule - Wikipedia

    en.wikipedia.org/wiki/Cramer's_rule

    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 ...

  7. Conjugate transpose - Wikipedia

    en.wikipedia.org/wiki/Conjugate_transpose

    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) .

  8. Matrix decomposition - Wikipedia

    en.wikipedia.org/wiki/Matrix_decomposition

    Matrix decomposition. In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.

  9. Incomplete LU factorization - Wikipedia

    en.wikipedia.org/wiki/Incomplete_LU_factorization

    A common choice is to use the sparsity pattern of A 2 instead of A; this matrix is appreciably more dense than A, but still sparse over all. This preconditioner is called ILU(1). One can then generalize this procedure; the ILU(k) preconditioner of a matrix A is the incomplete LU factorization with the sparsity pattern of the matrix A k+1.