Schurfaktoriseringer

Schurfaktoriseringer, also known in English as Schur factorization or Schur decomposition, is a fundamental concept in linear algebra. It states that every square matrix A can be expressed in the form A = Q T Q*, where Q is a unitary (or orthogonal in the real case) matrix and T is upper triangular. In the real case, T is often described as upper quasi-triangular, containing 2x2 blocks to represent complex conjugate eigenvalue pairs.

Existence and interpretation: For any complex n-by-n matrix A, there exists a unitary Q such that Q* A Q = T is upper triangular. The eigenvalues of A appear on the diagonal of T, equal to Tii. The Schur form is a compact, numerically stable representation of A that preserves eigenvalues under unitary similarity.

Computation and stability: The Schur decomposition is typically obtained as a byproduct of the QR algorithm, a staple method in numerical linear algebra for eigenvalue computations. The process is numerically stable because unitary transformations preserve norms and conditioning properties. In practice, the decomposition provides a reliable basis for further computations, such as evaluating functions of matrices.

Applications: Schurfaktoriseringer is used for computing eigenvalues, solving matrix equations, and evaluating matrix functions. It also underpins methods in control theory, signal processing, and numerical solutions to differential equations, where stable spectral information is essential.

Relation to other forms: The Schur decomposition is more general than diagonalization; every matrix has a Schur form even if it is not diagonalizable. If A is normal, the Schur form can be chosen diagonal, with Q unitary. The decomposition thus provides a robust framework for understanding the spectrum and behavior of matrices.