We characterize optimal rank-1 matrix approximations with Hankel or Toeplitz structure with regard to two different norms, the Frobenius norm and the spectral norm, in a new way. More precisely, we show that these rank-1 matrix approximation problems can be solved by maximizing special rational functions. Our approach enables us to show that the optimal solutions with respect to these two norms have completely different structure and only coincide in the trivial case when the singular value decomposition already provides an optimal rank-1 approximation with the desired Hankel or Toeplitz structure. We also prove that the Cadzow algorithm for structured low-rank approximations always converges to a fixed point in the rank-1 case. However, it usually does not converge to the optimal solution, neither with regard to the Frobenius norm nor the spectral norm.

我们以一种新的方式用 Hankel 或 Toeplitz 结构表征了关于两个不同范数（Frobenius 范数和谱范数）的最优秩 1 矩阵近似。更准确地说，我们表明这些秩为 1 的矩阵逼近问题可以通过最大化特殊有理函数来解决。我们的方法使我们能够证明关于这两个范数的最佳解决方案具有完全不同的结构，并且仅在奇异值分解已经提供具有所需 Hankel 或 Toeplitz 结构的最佳秩 1 近似值的平凡情况下才一致。我们还证明了用于结构化低秩近似的 Cadzow 算法在秩为 1 的情况下总是收敛到一个固定点。然而，它通常不会收敛到最优解，