Algebra — page 5
The nearest point is not the best guess
Measure ten unrelated quantities once each, with noise, and report the measurements. Nothing seems better. Yet pulling all ten towards nought by an amount the data choose gives a smaller total error whatever the true values are — in three dimensions or more, and never in one or two. The reason is a right angle: in many dimensions noise is nearly perpendicular to the truth, so a measurement is always too long, and the best correction is the shadow of the truth on it.
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.