If $\mathbf{A} \in \mathbb{R}^{nxm}$ is non-negative, is $\mathbf{A}^T\mathbf{A}$ also non-negative? Is there an associated lemma/theorem?
2026-04-03 03:16:49.1775186209
Does non-negative $\mathbf{A}$ imply non-negative $\mathbf{A}^T\mathbf{A}$?
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In the sense of positivity for matrices and operators, this is true even if $A$ is not non-negative since for any vector $x\in\mathbb{R}^m$ $$ x\cdot (A^TA) · x = (Ax)\cdot(Ax) = |Ax|^2 ≥ 0 $$