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Let $A$ be a $0/1$ square matrix which can be permuted to a non singular or a singular lower triangular matrix. Determinant is either $0$ or $1$. Can we provide tighter upper bounds on its spectral norm better than the usual bounds?

Assume the matrix has no $2\times2$ bicliques.

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    $\begingroup$ Any matrix can be written as a sum of an upper and a lower triangular matrices. So, up to a factor $2$, there cannot be fundamentally better bounds for triangular matrices. $\endgroup$ Commented Oct 31, 2021 at 16:18
  • $\begingroup$ Saying that the determinant/permanent is $0$ or $1$ is not a very strong constraint on a triangular matrix ;). $\endgroup$ Commented Nov 1, 2021 at 6:55
  • $\begingroup$ Sure, by the argument in my first comment, $\geq n/2$ is possible (and is attained on the matrix with $1$ everywhere above the diagonal). Of course, now that the question has changed, my comments are less relevant. $\endgroup$ Commented Nov 1, 2021 at 7:10
  • $\begingroup$ If you post an answer I will accept with $n/2$ solution. $\endgroup$ Commented Nov 1, 2021 at 7:40

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