Let $A\in\mathbb{C}^{m\times n}$ be a complex matrix. Let $B_k$ be a best rank-$k$ approximation to $A$ such that \begin{equation*} B_k\in\arg\min\limits_{{\rm rank}(B)=k}||A-B||_F, \end{equation*} where $||\cdot||_F$ denotes the Frobenius norm. Then, I guess $k$ largest singular values of $B_k$ is identical with the $k$ largest singular values of $A$ and other singular values of $B_k$ are all zero. Is my conjecture correct? Can anyone help me prove this?
2026-03-25 17:39:13.1774460353
The singular values of the best rank-$k$ approximation to a matrix
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