Given a matrix $X$, its spectral norm smaller than and equal to one, $\|X\|_2\leq 1$. can we say that the matrix have column of $\ell_2$ norm smaller than and equal to one? i.e., $\|X_i\|_2\leq 1$? where $X_i$ denote the $i$-th column of $X$.
2026-04-09 16:59:08.1775753948
Given a matrix, its spectral norm smaller than one, can we say that the matrix have column of L2 norm smaller than one?
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$$ \forall i, \ \ \ 1\geq\|X\|_2 = \max_{\|v\|_2=1}\|Xv\|\geq\|Xe_i\|_2 = \|X_i\|_2. $$