Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Rank deficiency — where it appears

The shortfall between a matrix's rank and its smaller dimension, which is the dimension of its null space. It is exact when the matrix is built from integers and is otherwise a judgement, since a decomposition returns small singular values where the mathematics has zeros and something has to decide which of them count as zero.

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

1234567891010⁻¹⁸10⁻¹⁵10⁻¹²10⁻⁹10⁻⁶10⁻³1indexsingular valuecutoff, σ₁ · 10⁻¹⁰numerical rank 10gap 8.2·10⁶an opiniontrue rank 410×10, built with 4 nonzero valuesrank is a decision

Rank is a decision

A floating-point matrix does not have a rank. It has a spectrum of singular values, and somewhere in that spectrum is a place where the values stop being signal and start being noise. Deciding where is a judgement, and the evidence for it is a gap.

spectra · Rank σ14.87σ22.55σ31.08σ40.782σ58.04·10⁻¹⁷σ610⁻¹⁸σ710⁻¹⁸n = 7, and det(A − λB) has degree 4an integer, and a judgementdegree of det(A − λB), exactly4infinite eigenvalues, from the degree3singular values below the cut3largest gap in the spectrum∞a degree cannot be nearly threeand a singular value can be nearly zero

The largest gap is inside the null space

The rule recommended for counting a pencil's infinite eigenvalues is to cut at the largest gap in the singular values of B. On integer pencils, with no perturbation anywhere and an exact answer available from the characteristic polynomial, it returns the wrong count on nine of twenty-five — because the singular values that are mathematically zero come back spread over a hundred and forty orders of magnitude, and the largest ratio in the list is between two of them.

spectra · Pencil

Named alongside it

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

Numerical rankSingular valuesColumn pivotingCondition numberDescriptor systemDeterminantExact arithmeticInfinite eigenvalueJacobi's eigenvalue methodLargest gap ruleMatrix pencilNull-space

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