Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Polar decomposition — where it appears

Writing a matrix as an orthogonal factor times a symmetric positive semidefinite one — a rotation after a stretch along orthogonal axes.

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

how far is it from A to an orthogonal matrix?smaller is nearer · the polar factor minimises this in every unitarily invariant normpolar factor U1.8554QR, signs fixed2.1265QR as returned3.8226200 drawn at randomκ = 10polar factor1.9QR, signs fixed2.1QR as returned3.8best of 200 random2.7‖A − QR‖ is the same either wayand ‖A − Q‖ is not

The nearest orthogonal matrix

Every field that has to clean up a drifted rotation reaches for QR, and QR does not answer the question. The nearest orthogonal matrix is the orthogonal factor of the polar decomposition — nearer by about a tenth, and, more to the point, the same matrix whatever order the columns were written in. QR's answer changes completely.

orthogonality · polar decomposition 159131721252910⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹steprelative error in the orthogonal factorNewtonNewton, scaledNewton–Schulzone fixed point, three costsscaled Newton, steps7Newton–Schulz, steps28Newton at step 657scaled Newton at step 64.4·10⁻¹³a Newton step needs an inverseand a Schulz step needs two products

An iteration that only multiplies

Newton's iteration for the polar factor needs an inverse every step. Newton–Schulz needs only matrix products — nothing that reads an entry, nothing that pivots — and it converges if and only if every singular value is below √3. At 1.73205 it converges and at 1.73206 it returns an orthogonal matrix that is not the answer, with a residual of 5·10⁻¹⁶ and nothing to say so.

orthogonality · polar decomposition

Named alongside it

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

OrthogonalityCondition numberConvergence rateFixed pointFlop countGram–SchmidtHouseholderInvarianceMatrix-freeNewton iterationQR factorisationRotation

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