QRdekompozíció

QRdekompozíció, also known as QR factorization, is a fundamental matrix decomposition technique in linear algebra. It decomposes a given matrix A into the product of an orthogonal matrix Q and an upper triangular matrix R. Mathematically, this is expressed as A = QR.

The matrix Q is an orthogonal matrix, meaning its columns are orthonormal vectors. This property implies that Q^T Q = I, where Q^T is the transpose of Q and I is the identity matrix. The matrix R is an upper triangular matrix, characterized by having all entries below the main diagonal equal to zero.

QR decomposition is widely used in numerical analysis and scientific computing. One of its primary applications is in solving systems of linear equations. If we have a system Ax = b, we can substitute A with QR, leading to QRx = b. Multiplying both sides by Q^T (since Q is orthogonal, Q^T is its inverse), we get Rx = Q^T b. This transformed system is much easier to solve for x because R is an upper triangular matrix, allowing for efficient back-substitution.

Another significant application of QR decomposition is in the computation of eigenvalues and eigenvectors, particularly through the QR algorithm. It also finds use in least squares problems, where it provides a numerically stable way to find the solution that minimizes the error. The decomposition can be achieved using various methods, including the Gram-Schmidt process, Householder reflections, and Givens rotations, each offering different trade-offs in terms of numerical stability and computational cost.