Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Hyperdeterminant — where it appears

A polynomial in the entries of a 2 x 2 x 2 tensor whose sign decides its rank exactly. It is the discriminant of the determinant of the pencil formed by the tensor's two slices, so a rank here is computed by four multiplications and a subtraction.

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

110¹10²10³10⁻³10⁻²10⁻¹110¹10²10³n‖A_n − A‖ and the largest term's norm‖A_n − A‖the larger of its two termsan infimum that is not attainedn1024‖A_n − A‖0.0017largest term1024their product1.7√31.7the distance goes to zeroand nothing reaches it

A nearest point that is not there

Eckart and Young guarantee that a matrix has a best rank-k approximation and that the truncated SVD is it. For three indices the guarantee is false in the strongest available way — there are tensors whose distance to the rank-two set is zero and which no rank-two tensor equals.

tensor · tensor rank 10¹10²10³10⁴00.050.10.150.20.250.30.350.40.450.50.550.60.650.70.750.80.850.90.951drawsshare with real rank twoπ/4 = 0.78539815,705 of 19,953 have rank twoa probability with a closed formdraws2·10⁴rank two1.6·10⁴share0.79π/40.79standard errors out0.59two typical ranksand the split is π/4

A rank that is not a property of the tensor

The same eight real numbers have rank three over the reals and rank two over the complexes, and a random 2 × 2 × 2 tensor has rank two with probability exactly π/4. Neither sentence has an analogue for matrices, where the rank is one number and a random matrix has the largest one.

tensor · tensor rank

Named alongside it

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

Border rankEckart–YoungLow-rank approximationNumerical rankTensor rankUnfoldingExact ground truthIll posed problemSeeded generatorSeparabilityTruncated svdTypical rank

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