A = LU. The matrix U is found by an upper triangularization procedure which involves left-multiplying A by a series of matrices to form the product , so that equivalently . Eigenvalue decomposition The eigenvalue decomposition of a matrix is , where the columns of X are the eigenvectors of A, and is a diagonal matrix the diagonal entries of which are the corresponding eigenvalues of A. There is no direct method for finding the eigenvalue decomposition of an arbitrary matrix. Because it is not possible to write a program that finds the exact roots of an arbitrary polynomial in finite time, any general eigenvalue solver must necessarily be iterative. Algorithms Gaussian elimination From the numerical linear algebra perspective, Gaussian elimination is a procedure for factoring a matrix A into its LU factorization, which Gaussian elimination accomplishes by left-multiplying A by a succession of matrices until U is upper triangular and L is lower triangular, where . Naive programs for Gaussian elimination are notoriously highly unstable, and produce huge errors when applied to matrices with many significant digits. The simplest solution is to introduce pivoting, which produces a modified Gaussian elimination algorithm that is stable. Solutions of linear systems Numerical linear algebra characteristically approaches matrices as a concatenation of columns vectors. In order to solve the linear system , the traditional algebraic approach is to understand x as the product of with b. Numerical linear algebra instead interprets x as the vector of coefficients of the linear expansion of b in the basis formed by the columns of A. Many different decompositions can be used to solve the linear problem, depending on the characteristics of the matrix A and the vectors x and b, which may make one factorization much easier to obtain than others. If A = QR is a QR factorization of A, then equivalently . This is easy to compute as a matrix factorization. If is an eigendecomposition A, and we seek to find b so that b = Ax, with and , then we have . This is closely related to the solution to the linear system using the singular value decomposition, because singular values of a matrix are the square roots of its eigenvalues. And if A = LU is an LU factorization of A, then Ax = b can be solved using the triangular matrices Ly = b and Ux = y. Least squares