Schurdecompositie

Schurdecompositie, also known as Schur decomposition or Schur factorization, is a fundamental concept in linear algebra. It provides a way to transform a given square matrix into an upper triangular matrix by applying a unitary similarity transformation. Specifically, for any complex square matrix A, there exists a unitary matrix U and an upper triangular matrix T such that A = U T U*, where U* denotes the conjugate transpose of U. For real matrices, the decomposition can be slightly different, involving a real orthogonal matrix Q and a real upper quasi-triangular matrix T. This quasi-triangular form has 1x1 or 2x2 real blocks on the diagonal.

The Schur decomposition is particularly useful because it preserves the eigenvalues of the original matrix. The diagonal entries of the upper triangular matrix T (or the eigenvalues of the 2x2 blocks in the real case) are precisely the eigenvalues of A, including their algebraic multiplicities. This property makes the Schur decomposition a powerful tool for eigenvalue computation and analysis. It forms the basis for many algorithms used to solve linear algebra problems, such as finding eigenvalues and eigenvectors.

The existence of the Schur decomposition is guaranteed for any square matrix. The unitary matrix U essentially represents a change of basis that reveals the triangular structure of the matrix. This decomposition is not unique; if T is a Schur form of A, then any permutation of the diagonal elements of T, corresponding to a permutation of the columns and rows of U, will also result in a Schur form. Despite this lack of uniqueness, the set of eigenvalues remains invariant.