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. In numerical analysis, different decompositions are used to implement efficient matrix algorithms. For instance, when solving a system of linear equations , the matrix A can be decomposed via the LU decomposition. The LU decomposition factorizes a matrix into a lower triangular matrix L and an upper triangular matrix U. The systems and require fewer additions and multiplications to solve, compared with the original system , though one might require significantly more digits in inexact arithmetic such as floating point. Similarly, the QR decomposition expresses A as QR with Q an orthogonal matrix and R an upper triangular matrix. The system Q(Rx) = b is solved by Rx = QTb = c, and the system Rx = c is solved by 'back substitution'. The number of additions and multiplications required is about twice that of using the LU solver, but no more digits are required in inexact arithmetic because the QR decomposition is numerically stable. LU decomposition Traditionally applicable to: square matrix A, although rectangular matrices can be applicable. Decomposition: , where L is lower triangular and U is upper triangular Related: the LDU decomposition is , where L is lower triangular with ones on the diagonal, U is upper triangular with ones on the diagonal, and D is a diagonal matrix. Related: the LUP decomposition is , where L is lower triangular, U is upper triangular, and P is a permutation matrix. Existence: An LUP decomposition exists for any square matrix A. When P is an identity matrix, the LUP decomposition reduces to the LU decomposition. Comments: The LUP and LU decompositions are useful in solving an n-by-n system of linear equations . These decompositions summarize the process of Gaussian elimination in matrix form.