A Schur decomposition of a matrix involves factorizing it into a unitary matrix and an upper triangular matrix, with eigenvalues appearing on the diagonal. The columns of the unitary matrix are known as Schur vectors and span an invariant subspace. Eigenvectors can be computed by solving upper triangular systems. The Schur decomposition is useful for computing matrix functions and offers advantages over the Jordan canonical form. Real matrices have a real Schur decomposition, and the process can be computed efficiently using the QR algorithm. MATLAB has functions for calculating the Schur decomposition. Van Loan emphasizes the superiority of the Schur decomposition over the Jordan decomposition.
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