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Concept

Gram schmidt

The procedure that turns any independent set of vectors into a perpendicular one by subtracting from each vector its shadows on those already built. It preserves the space the vectors span, it reports dependence by returning a vector of length zero, and the coefficients it discards form the triangular half of a QR factorisation.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

One subtraction clears a direction. Gram–Schmidt on two planar vectors, in 3 panels: the pair as given, the shadow of the second on the first, and the perpendicular pair that is left when the shadow is removed.

One subtraction clears a direction

A basis is a set of directions to measure along, and most bases are awkward because the directions get in each other's way. Removing one shadow at a time turns any basis into one where every coordinate is a shadow and nothing interferes.

algebra · Inner product The nearest point of the plane the columns span. A target vector in space, the plane spanned by two columns, the point of that plane nearest the target, and the residual joining them, which meets the plane at a right angle.

The nearest point of a flat thing

More equations than unknowns almost never have a solution. Asking instead for the point of a plane nearest to where the answer should have been turns an unanswerable question into a shadow, and the shadow is what a line of best fit is.

algebra · Inner product A dimension of 1.2576, from two stretching rates. The running averages of the Hénon map's two Lyapunov exponents, settling at 0.4177 and -1.6217. Kaplan and Yorke's formula turns them into a dimension of 1.2576 without counting a single box.

A dimension from the stretching rates

An attractor has no construction rule, so its dimension has to be counted — that was the rung below's argument for defining dimension by counting at all. Kaplan and Yorke's formula computes it instead, from two numbers that describe the map and never look at the set.

dynamics · Fractal dimension

Named alongside it

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

ApproximationBasisDot productInner productLeast squaresLinear independenceOrthogonalityOrthonormal basisProjectionResidualSubspaceAttractor

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