| 1. | A orthogonal or unitary matrix, and a triangular matrix.
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| 2. | A triangular matrix is one that is either lower triangular or upper triangular.
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| 3. | The determinant of a triangular matrix equals the product of the diagonal entries.
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| 4. | The product of a Hessenberg matrix with a triangular matrix is again Hessenberg.
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| 5. | Let be a normal upper triangular matrix.
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| 6. | Hence, the lower triangular matrix " L " we are looking for is calculated as
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| 7. | Then by Schur decomposition it is unitary similar to an upper-triangular matrix, say,.
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| 8. | It results in a " unit lower triangular " matrix and an upper triangular matrix.
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| 9. | The eigenvalues of a triangular matrix are listed on the diagonal, and the eigenvalue problem is solved.
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| 10. | Then " S " is an upper-triangular matrix with all diagonal entries being positive.
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