Hadamard growth problem: linear growth factor
Establish whether the Gaussian elimination growth factor ρ(H_n) equals n for every Hadamard matrix H_n. Determine if the maximal intermediate-element growth under Gaussian elimination (as defined in the paper) is exactly linear in n across the class of Hadamard matrices.
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A similarly famous open problem for numerical analysis involves the growth problem for Hadamard matrices, where it is believed $\rho(H_n) = n$. This has been established for all Sylvester Hadamard matrices (see ), while for general Hadamard matrices this has only been proven up to $n = 16$ . The growth problem for Hadamard matrices is a sub-question for the growth problem when restricted to orthogonal matrices. For instance, the orthogonal growth problem remains open in GEPP even, which was recently explored in .