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Hadamard matrix

A square matrix of entries +1 and −1 whose rows are mutually perpendicular, so that its determinant is as large as any sign matrix of that size allows. Such matrices exist only in sizes 1, 2 and multiples of 4, and whether every multiple of 4 has one is unknown.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The same two lengths at 3 angles, and the area largest at the right angle. Parallelograms spanned by columns of lengths 1.6 and 1.25 at angles 38, 90, 142 degrees. Their areas are 1.23, 2.00, 1.23; the bound 2.00 is the product of the lengths and is reached only when the columns are perpendicular.

The biggest box built from signs

Fill a square table with plus and minus ones and ask how large its determinant can be. The columns all have the same length, so the answer is a box with fixed edges — largest when every corner is square, which is possible only when the size is a multiple of four.

algebra · Determinant How often a table of random signs is singular. n=2: 0.5000 (parallel 0.500); n=3: 0.6250 (parallel 0.625); n=4: 0.6602 (parallel 0.660); n=5: 0.6558 (parallel 0.628); n=6: 0.6270 (parallel 0.547); n=7: 0.5810 (parallel 0.442); n=8: 0.5216 (parallel 0.329); n=9: 0.4515 (parallel 0.234); n=10: 0.3757 (parallel 0.154); n=11: 0.2963 (parallel 0.0978); n=12: 0.2274 (parallel 0.0619); n=13: 0.1639 (parallel 0.0361); n=14: 0.1136 (parallel 0.0217); n=15: 0.07550 (parallel 0.0127); n=16: 0.04668 (parallel 0.00683); n=18: 0.01639 (parallel 0.00229); n=20: 0.004725 (parallel 0.000683)

How often a table of random signs is flat

Fill a square table with random plus and minus signs and ask whether its determinant is nought. For four rows the answer is yes two times in three. The chance then falls, and since 2020 it has been known to fall at exactly the rate of the dullest accident, two rows or columns equal or opposite. The conjecture is that this accident is, in the end, the whole story. Every size a computer can reach says otherwise, and the theory says the computer has not gone far enough.

algebra · Determinant

Named alongside it

The objects these essays reach for when they reach for this one.

DeterminantMatrixBoundError-correcting codeExhaustive searchLinear independenceMonte CarloOrthogonalityQuadratic residueRandom matrixVolume

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