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The Lanczos-Ritz values appearing in a reduction to an upper triangular semiseparable matrix

We know from the reduction algorithm as presented in Theorem 51, that the intermediate matrices $ A^{(k)}$, have the upper $ k\times n$ matrix $ S^u_k$ of upper triangular semiseparable form. Using the relations provided in Theorem 58, we know that the singular values of this matrix $ S^u_k$ are the square roots of the eigenvalues of the matrix $ \tilde{S}_k=S^u_k {S^u_k}^T$, and the eigenvalues of the matrix $ \tilde{S}_k$ as can be seen when combining Theorem 58 and the results from Section 5.2.1 are the Lanczos-Ritz values.

Raf Vandebril 2004-05-03