QRdekompozíciót

QR dekompozíció, also known as QR decomposition, is a matrix factorization technique used in linear algebra. It decomposes a given matrix into a product of an orthogonal matrix and an upper triangular matrix. The process is particularly useful in numerical analysis and various applications in science and engineering.

The QR decomposition of a matrix A can be written as A = QR, where Q is an orthogonal matrix (Q^T Q = I, where I is the identity matrix) and R is an upper triangular matrix. This decomposition is often used to solve systems of linear equations, find eigenvalues, and perform least squares fitting.

The QR decomposition can be computed using several algorithms, including the Gram-Schmidt process and the Householder transformations. The choice of algorithm depends on the specific requirements of the application, such as numerical stability and computational efficiency.

One of the key advantages of QR decomposition is its numerical stability. The orthogonal matrix Q preserves the Euclidean norm of vectors, which helps in maintaining numerical accuracy during computations. This property makes QR decomposition a preferred method for solving ill-conditioned systems of equations.

In summary, QR decomposition is a fundamental tool in linear algebra with wide-ranging applications. Its ability to decompose a matrix into an orthogonal and an upper triangular component, along with its numerical stability, makes it a valuable technique in various fields of science and engineering.