Schur Decomposition
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High Quality Content by WIKIPEDIA articles! In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is an important matrix decomposition. A constructive proof for the Schur decomposition is as follows: every operator A on a complex finite-dimensional vector space has an eigenvalue , corresponding to some eigenspace V . Let V be its orthogonal complement. It is clear that, with respect to this orthogonal decomposition, A has matrix representation (one can pick here any orthonormal bases spanning V and V respectively) A = begin{b...
High Quality Content by WIKIPEDIA articles! In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is an important matrix decomposition. A constructive proof for the Schur decomposition is as follows: every operator A on a complex finite-dimensional vector space has an eigenvalue , corresponding to some eigenspace V . Let V be its orthogonal complement. It is clear that, with respect to this orthogonal decomposition, A has matrix representation (one can pick here any orthonormal bases spanning V and V respectively) A = begin{bmatrix} lambda , I_{lambda} & A_{12} 0 & A_{22} end{bmatrix}: begin{matrix} V_{lambda} oplus V_{lambda}^{perp} end{matrix} rightarrow begin{matrix} V_{lambda} oplus V_{lambda}^{perp} end{matrix} where I is the identity operator on V . The above matrix would be upper-triangular except for the A22 block. But exactly the same procedure can be applied to the sub-matrix A22, viewed as an operatoron V , and its submatrices. Continue this way n times. Thus the space Cn will be exhausted and the procedure has yielded the desired result.
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