= Hadamard determinant inequality
{c}
{title2=$|\det A|\leq\prod_i|r_i|$}
= Hadamard inequality
{c}
{synonym}
For a square <matrix> with row vectors $r_i$, the absolute <determinant> is at most the product of the <Euclidean norms> of its rows. If the rows are dependent the <determinant> is zero. Otherwise <Gram-Schmidt process> leaves the <determinant> unchanged and replaces each row by its component orthogonal to its predecessors. The resulting orthogonal rows have lengths no greater than the original rows, and their volume is their product. This proves the inequality; equality holds for mutually orthogonal rows.
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