QRfaktoriseringar

QR factorization, also known as QR decomposition, is a matrix factorization technique used in linear algebra and numerical analysis. It decomposes a given matrix A into the product of an orthogonal matrix Q and an upper triangular matrix R. This factorization is particularly useful in solving linear least squares problems, eigenvalue problems, and in various numerical algorithms.

The orthogonal matrix Q has the property that its columns are orthonormal vectors, meaning that the dot product of any two distinct columns is zero, and the dot product of a column with itself is one. The upper triangular matrix R has non-zero elements only on and above the main diagonal.

The QR factorization can be computed using several algorithms, including the Gram-Schmidt process, Householder transformations, and Givens rotations. Each method has its own advantages and is chosen based on the specific requirements of the problem at hand.

One of the primary applications of QR factorization is in solving linear least squares problems. Given a system of linear equations Ax = b, where A is an m-by-n matrix and b is a vector of length m, the QR factorization can be used to find the solution x that minimizes the residual ||Ax - b||. This is achieved by solving the triangular system Rx = Q^T b.

QR factorization is also used in the computation of eigenvalues and eigenvectors of a matrix. The QR algorithm, for example, iteratively applies QR factorization to a matrix to find its eigenvalues.

In summary, QR factorization is a powerful tool in linear algebra with wide-ranging applications in numerical analysis and scientific computing. Its ability to decompose a matrix into an orthogonal and an upper triangular component makes it a versatile method for solving various types of linear problems.