Numerical Linear Algebra
where the algebra stops being the arithmetic
Generator

Reversing the columns of a 6×6 matrix: what each orthogonalisation returns

One function in the polar library, called 7 times across 2 essays. Below: what it draws at its defaults, what it draws at every value an essay asks for, the 12 claims it put to the test while drawing them, and where it stands against the rule this site is named for.

At its defaults it draws reversing the columns of a 6×6 matrix: what each orthogonalisation returns. Both grids print the entries of Q₁P − Q₂, where Q₁ orthogonalises A and Q₂ orthogonalises A with its columns reversed. If the answer permuted with the columns, both grids would be zero. The polar factor's is, in the Frobenius norm: ‖U₁P − U₂‖ = 1.53·10⁻¹⁵. QR's is not: ‖Q₁P − Q₂‖ = 2.353 on matrices whose own Frobenius norm is 2.449, so the two answers are essentially unrelated. Gram–Schmidt and Householder both build Q one column at a time, and the first column is treated differently from the last.

orthogonalisation-order is one function in lib/figures/polar.js — the polar factor — the nearest orthogonal matrix, and two ways to it without an svd. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.

Reversing the columns of a 6×6 matrix: what each orthogonalisation returnsBoth grids print the entries of Q₁P − Q₂, where Q₁ orthogonalises A and Q₂ orthogonalises A with its columns reversed. If the answer permuted with the columns, both grids would be zero. The polar factor's is, in the Frobenius norm: ‖U₁P − U₂‖ = 1.53·10⁻¹⁵. QR's is not: ‖Q₁P − Q₂‖ = 2.353 on matrices whose own Frobenius norm is 2.449, so the two answers are essentially unrelated. Gram–Schmidt and Householder both build Q one column at a time, and the first column is treated differently from the last.polar: U₁P − U₂QR: Q₁P − Q₂-5·10⁻¹⁶3.3·10⁻¹⁶-2.2·10⁻¹⁶-1.7·10⁻¹⁶5.6·10⁻¹⁷-1.9·10⁻¹⁶1.7·10⁻¹⁶6.7·10⁻¹⁶3.9·10⁻¹⁶1.1·10⁻¹⁶2.2·10⁻¹⁶5.6·10⁻¹⁶5.6·10⁻¹⁷005.6·10⁻¹⁷1.7·10⁻¹⁶-3.3·10⁻¹⁶4.2·10⁻¹⁷1.1·10⁻¹⁶-2.2·10⁻¹⁶-1.1·10⁻¹⁶2.2·10⁻¹⁶-2.6·10⁻¹⁶-3.3·10⁻¹⁶-4.4·10⁻¹⁶-1.1·10⁻¹⁶5.6·10⁻¹⁷0-3.3·10⁻¹⁶1.7·10⁻¹⁶1.1·10⁻¹⁶1.1·10⁻¹⁶1.1·10⁻¹⁶01.7·10⁻¹⁶0.56-0.15-0.21-0.330.06-0.64-0.430.310.310.120.20.73-0.130.64-0.031-0.0028-0.420.075-0.18-0.43-0.130.21-0.29-0.420.640.630.670.62-0.52-0.420.26-0.520.022-0.0870.2-0.24Frobenius norms, columns reordered‖U₁P − U₂‖1.5·10⁻¹⁵‖Q₁P − Q₂‖2.4‖Q‖, for scale2.4κ of the matrix10Frobenius distance between the two answers, on one scale0 to 4polar1.5·10⁻¹⁵QR2.353‖Q‖ = 2.449the column space did not moveand one of the two answers did

Both grids print the entries of Q₁P − Q₂, where Q₁ orthogonalises A and Q₂ orthogonalises A with its columns reversed. If the answer permuted with the columns, both grids would be zero. The polar factor's is, in the Frobenius norm: ‖U₁P − U₂‖ = 1.53·10⁻¹⁵. QR's is not: ‖Q₁P − Q₂‖ = 2.353 on matrices whose own Frobenius norm is 2.449, so the two answers are essentially unrelated. Gram–Schmidt and Householder both build Q one column at a time, and the first column is treated differently from the last.

logKappa: 2

The arguments are the ones A test with no answer in it passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Reversing the columns of a 6×6 matrix: what each orthogonalisation returnsBoth grids print the entries of Q₁P − Q₂, where Q₁ orthogonalises A and Q₂ orthogonalises A with its columns reversed. If the answer permuted with the columns, both grids would be zero. The polar factor's is, in the Frobenius norm: ‖U₁P − U₂‖ = 2.78·10⁻¹⁵. QR's is not: ‖Q₁P − Q₂‖ = 3.048 on matrices whose own Frobenius norm is 2.449, so the two answers are essentially unrelated. Gram–Schmidt and Householder both build Q one column at a time, and the first column is treated differently from the last.polar: U₁P − U₂QR: Q₁P − Q₂-3.3·10⁻¹⁶-3.3·10⁻¹⁶2.2·10⁻¹⁶-1.9·10⁻¹⁶5.6·10⁻¹⁷-1.7·10⁻¹⁶6.7·10⁻¹⁶-8.9·10⁻¹⁶2.8·10⁻¹⁶-5.6·10⁻¹⁶-3.6·10⁻¹⁶6.1·10⁻¹⁶-8.3·10⁻¹⁶8.3·10⁻¹⁶-7.2·10⁻¹⁶-10·10⁻¹⁶2.2·10⁻¹⁶4.4·10⁻¹⁶1.7·10⁻¹⁶-4.4·10⁻¹⁶-5.6·10⁻¹⁷5.6·10⁻¹⁷0-7.2·10⁻¹⁶-4.4·10⁻¹⁶3.9·10⁻¹⁶5.6·10⁻¹⁷2.8·10⁻¹⁶-1.1·10⁻¹⁶-3.3·10⁻¹⁶2.2·10⁻¹⁶1.1·10⁻¹⁶-2.8·10⁻¹⁶5.6·10⁻¹⁶-2.8·10⁻¹⁶5.6·10⁻¹⁶0.67-0.35-0.23-0.620.19-0.72-0.770.170.20.290.190.93-0.180.55-0.0290.06-0.460.11-0.17-0.61-0.230.38-0.55-0.380.750.870.661.1-0.36-0.530.42-0.670.180.180.52-0.3Frobenius norms, columns reordered‖U₁P − U₂‖2.8·10⁻¹⁵‖Q₁P − Q₂‖3‖Q‖, for scale2.4κ of the matrix100Frobenius distance between the two answers, on one scale0 to 4polar2.8·10⁻¹⁵QR3.048‖Q‖ = 2.449the column space did not moveand one of the two answers did

Both grids print the entries of Q₁P − Q₂, where Q₁ orthogonalises A and Q₂ orthogonalises A with its columns reversed. If the answer permuted with the columns, both grids would be zero. The polar factor's is, in the Frobenius norm: ‖U₁P − U₂‖ = 2.78·10⁻¹⁵. QR's is not: ‖Q₁P − Q₂‖ = 3.048 on matrices whose own Frobenius norm is 2.449, so the two answers are essentially unrelated. Gram–Schmidt and Householder both build Q one column at a time, and the first column is treated differently from the last.

logKappa: 1

The arguments are the ones A test with no answer in it passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Reversing the columns of a 6×6 matrix: what each orthogonalisation returnsBoth grids print the entries of Q₁P − Q₂, where Q₁ orthogonalises A and Q₂ orthogonalises A with its columns reversed. If the answer permuted with the columns, both grids would be zero. The polar factor's is, in the Frobenius norm: ‖U₁P − U₂‖ = 1.53·10⁻¹⁵. QR's is not: ‖Q₁P − Q₂‖ = 2.353 on matrices whose own Frobenius norm is 2.449, so the two answers are essentially unrelated. Gram–Schmidt and Householder both build Q one column at a time, and the first column is treated differently from the last.polar: U₁P − U₂QR: Q₁P − Q₂-5·10⁻¹⁶3.3·10⁻¹⁶-2.2·10⁻¹⁶-1.7·10⁻¹⁶5.6·10⁻¹⁷-1.9·10⁻¹⁶1.7·10⁻¹⁶6.7·10⁻¹⁶3.9·10⁻¹⁶1.1·10⁻¹⁶2.2·10⁻¹⁶5.6·10⁻¹⁶5.6·10⁻¹⁷005.6·10⁻¹⁷1.7·10⁻¹⁶-3.3·10⁻¹⁶4.2·10⁻¹⁷1.1·10⁻¹⁶-2.2·10⁻¹⁶-1.1·10⁻¹⁶2.2·10⁻¹⁶-2.6·10⁻¹⁶-3.3·10⁻¹⁶-4.4·10⁻¹⁶-1.1·10⁻¹⁶5.6·10⁻¹⁷0-3.3·10⁻¹⁶1.7·10⁻¹⁶1.1·10⁻¹⁶1.1·10⁻¹⁶1.1·10⁻¹⁶01.7·10⁻¹⁶0.56-0.15-0.21-0.330.06-0.64-0.430.310.310.120.20.73-0.130.64-0.031-0.0028-0.420.075-0.18-0.43-0.130.21-0.29-0.420.640.630.670.62-0.52-0.420.26-0.520.022-0.0870.2-0.24Frobenius norms, columns reordered‖U₁P − U₂‖1.5·10⁻¹⁵‖Q₁P − Q₂‖2.4‖Q‖, for scale2.4κ of the matrix10Frobenius distance between the two answers, on one scale0 to 4polar1.5·10⁻¹⁵QR2.353‖Q‖ = 2.449the column space did not moveand one of the two answers did

Both grids print the entries of Q₁P − Q₂, where Q₁ orthogonalises A and Q₂ orthogonalises A with its columns reversed. If the answer permuted with the columns, both grids would be zero. The polar factor's is, in the Frobenius norm: ‖U₁P − U₂‖ = 1.53·10⁻¹⁵. QR's is not: ‖Q₁P − Q₂‖ = 2.353 on matrices whose own Frobenius norm is 2.449, so the two answers are essentially unrelated. Gram–Schmidt and Householder both build Q one column at a time, and the first column is treated differently from the last.

logKappa: 0

The arguments are the ones A test with no answer in it passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Reversing the columns of a 6×6 matrix: what each orthogonalisation returnsBoth grids print the entries of Q₁P − Q₂, where Q₁ orthogonalises A and Q₂ orthogonalises A with its columns reversed. If the answer permuted with the columns, both grids would be zero. The polar factor's is, in the Frobenius norm: ‖U₁P − U₂‖ = 1.8·10⁻¹⁵. QR's is not: ‖Q₁P − Q₂‖ = 0.070 on matrices whose own Frobenius norm is 2.449, so the two answers are essentially unrelated. Gram–Schmidt and Householder both build Q one column at a time, and the first column is treated differently from the last.polar: U₁P − U₂QR: Q₁P − Q₂6.7·10⁻¹⁶0-3.3·10⁻¹⁶5.6·10⁻¹⁷-3.9·10⁻¹⁶2.5·10⁻¹⁶3.3·10⁻¹⁶-4.4·10⁻¹⁶-1.1·10⁻¹⁶-1.7·10⁻¹⁶-3.1·10⁻¹⁶1.7·10⁻¹⁶1.1·10⁻¹⁶-2.2·10⁻¹⁶2.2·10⁻¹⁶-2.2·10⁻¹⁶2.8·10⁻¹⁶2.2·10⁻¹⁶-1.4·10⁻¹⁶5.6·10⁻¹⁷3.9·10⁻¹⁶1.1·10⁻¹⁶3.3·10⁻¹⁶2.5·10⁻¹⁶5.6·10⁻¹⁶2.2·10⁻¹⁶3.3·10⁻¹⁶3.3·10⁻¹⁶2.5·10⁻¹⁶-1.1·10⁻¹⁶-1.1·10⁻¹⁶-4.4·10⁻¹⁶6.7·10⁻¹⁶1.1·10⁻¹⁶-2.2·10⁻¹⁶-5.6·10⁻¹⁷0.015-7.8·10⁻⁴-0.007-0.0062-3.5·10⁻⁴-0.02-0.00890.0120.0150.00170.00560.021-0.00270.02-2.6·10⁻⁴0.0026-0.0160.0018-0.0051-0.015-0.00440.0034-0.0052-0.0150.0170.0150.0180.014-0.022-0.0130.0059-0.011-0.0012-0.00510.002-0.0073Frobenius norms, columns reordered‖U₁P − U₂‖1.8·10⁻¹⁵‖Q₁P − Q₂‖0.07‖Q‖, for scale2.4κ of the matrix1Frobenius distance between the two answers, on one scale0 to 4polar1.8·10⁻¹⁵QR0.070‖Q‖ = 2.449the column space did not moveand one of the two answers did

Both grids print the entries of Q₁P − Q₂, where Q₁ orthogonalises A and Q₂ orthogonalises A with its columns reversed. If the answer permuted with the columns, both grids would be zero. The polar factor's is, in the Frobenius norm: ‖U₁P − U₂‖ = 1.8·10⁻¹⁵. QR's is not: ‖Q₁P − Q₂‖ = 0.070 on matrices whose own Frobenius norm is 2.449, so the two answers are essentially unrelated. Gram–Schmidt and Householder both build Q one column at a time, and the first column is treated differently from the last.

n: 3

The arguments are the ones A test with no answer in it passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Reversing the columns of a 3×3 matrix: what each orthogonalisation returnsBoth grids print the entries of Q₁P − Q₂, where Q₁ orthogonalises A and Q₂ orthogonalises A with its columns reversed. If the answer permuted with the columns, both grids would be zero. The polar factor's is, in the Frobenius norm: ‖U₁P − U₂‖ = 2.5·10⁻¹⁶. QR's is not: ‖Q₁P − Q₂‖ = 1.190 on matrices whose own Frobenius norm is 1.732, so the two answers are essentially unrelated. Gram–Schmidt and Householder both build Q one column at a time, and the first column is treated differently from the last.polar: U₁P − U₂QR: Q₁P − Q₂5.6·10⁻¹⁷-1.7·10⁻¹⁶1.1·10⁻¹⁶-1.1·10⁻¹⁶2.8·10⁻¹⁷05.6·10⁻¹⁷05.6·10⁻¹⁷-0.24-0.65-0.32-0.12-0.32-0.13-0.16-0.310.77Frobenius norms, columns reordered‖U₁P − U₂‖2.5·10⁻¹⁶‖Q₁P − Q₂‖1.2‖Q‖, for scale1.7κ of the matrix10Frobenius distance between the two answers, on one scale0 to 4polar2.5·10⁻¹⁶QR1.190‖Q‖ = 1.732the column space did not moveand one of the two answers did

Both grids print the entries of Q₁P − Q₂, where Q₁ orthogonalises A and Q₂ orthogonalises A with its columns reversed. If the answer permuted with the columns, both grids would be zero. The polar factor's is, in the Frobenius norm: ‖U₁P − U₂‖ = 2.5·10⁻¹⁶. QR's is not: ‖Q₁P − Q₂‖ = 1.190 on matrices whose own Frobenius norm is 1.732, so the two answers are essentially unrelated. Gram–Schmidt and Householder both build Q one column at a time, and the first column is treated differently from the last.

n: 8, logKappa: 4

The arguments are the ones A test with no answer in it passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.

Reversing the columns of a 8×8 matrix: what each orthogonalisation returnsBoth grids print the entries of Q₁P − Q₂, where Q₁ orthogonalises A and Q₂ orthogonalises A with its columns reversed. If the answer permuted with the columns, both grids would be zero. The polar factor's is, in the Frobenius norm: ‖U₁P − U₂‖ = 2.45·10⁻¹⁴. QR's is not: ‖Q₁P − Q₂‖ = 3.955 on matrices whose own Frobenius norm is 2.828, so the two answers are essentially unrelated. Gram–Schmidt and Householder both build Q one column at a time, and the first column is treated differently from the last.polar: U₁P − U₂QR: Q₁P − Q₂1.5·10⁻¹⁵1.3·10⁻¹⁵-7.5·10⁻¹⁶-8.9·10⁻¹⁶5.6·10⁻¹⁷0-1.2·10⁻¹⁵5·10⁻¹⁶5.7·10⁻¹⁵8.9·10⁻¹⁶-5.4·10⁻¹⁶-2·10⁻¹⁵-2.1·10⁻¹⁵-1.6·10⁻¹⁵-4.4·10⁻¹⁵1.7·10⁻¹⁵-8.1·10⁻¹⁵9.5·10⁻¹⁵-3.7·10⁻¹⁵-5.7·10⁻¹⁵1.4·10⁻¹⁵-4.8·10⁻¹⁵-4.8·10⁻¹⁵7.2·10⁻¹⁶1.1·10⁻¹⁴-4.4·10⁻¹⁶-1.8·10⁻¹⁵-1.5·10⁻¹⁵-3.3·10⁻¹⁵-1.9·10⁻¹⁵-6.1·10⁻¹⁵2.3·10⁻¹⁵-4.7·10⁻¹⁶08.9·10⁻¹⁶-1.7·10⁻¹⁶-7.2·10⁻¹⁶-5.7·10⁻¹⁶1.1·10⁻¹⁵-1.7·10⁻¹⁶-4.7·10⁻¹⁶-3.1·10⁻¹⁵1.9·10⁻¹⁵1.8·10⁻¹⁵1.1·10⁻¹⁵1.9·10⁻¹⁵3.1·10⁻¹⁵-1.2·10⁻¹⁵-1.1·10⁻¹⁵-2.2·10⁻¹⁵6.1·10⁻¹⁶2.4·10⁻¹⁵1.6·10⁻¹⁵2.8·10⁻¹⁵2.6·10⁻¹⁵-1.2·10⁻¹⁵-2.9·10⁻¹⁵2.2·10⁻¹⁵-10·10⁻¹⁶-1.6·10⁻¹⁵1.8·10⁻¹⁵-8.9·10⁻¹⁶-4.4·10⁻¹⁶-3.9·10⁻¹⁶-0.220.730.37-0.120.22-10.510.15-0.180.490.420.26-0.470.770.51-0.069-0.43-0.78-0.390.190.730.290.0280.83-0.90.120.180.230.720.250.44-0.54-0.3-0.120.028-0.15-0.32-0.076-0.22-0.470.730.00580.370.31.3-0.34-0.71-0.13-0.470.44-0.370.0420.48-0.130.42-0.77-0.28-0.6310.17-0.390.24-0.73-0.4Frobenius norms, columns reordered‖U₁P − U₂‖2.4·10⁻¹⁴‖Q₁P − Q₂‖4‖Q‖, for scale2.8κ of the matrix10⁴Frobenius distance between the two answers, on one scale0 to 5polar2.4·10⁻¹⁴QR3.955‖Q‖ = 2.828the column space did not moveand one of the two answers did

Both grids print the entries of Q₁P − Q₂, where Q₁ orthogonalises A and Q₂ orthogonalises A with its columns reversed. If the answer permuted with the columns, both grids would be zero. The polar factor's is, in the Frobenius norm: ‖U₁P − U₂‖ = 2.45·10⁻¹⁴. QR's is not: ‖Q₁P − Q₂‖ = 3.955 on matrices whose own Frobenius norm is 2.828, so the two answers are essentially unrelated. Gram–Schmidt and Householder both build Q one column at a time, and the first column is treated differently from the last.

What it checked while drawing

Every figure above checked its own claims on the way to being drawn, and a claim that failed would have stopped the picture rather than shipped a wrong one. Those checks used to leave no trace at all: a passing one returned true and the only evidence the figure had checked anything was that nothing crashed. The list below is what they actually said, collected by running this generator with an observer installed — not a description of what it is believed to check.

12 distinct claims across 6 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.

the constructed matrix has κ = 10 — checked 4 times

a conditioning the construction covers

a power of ten rather than an exponent literal

a size a grid of differences can print

and QR's is a different matrix

by a fraction of its own size rather than a perturbation

matmul shapes agree

on a nearly orthogonal matrix QR's answer barely depends on the column order, which is the control

the polar factor of a column-permuted matrix is the permuted polar factor

Against the rule

The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.

Across the library: the rule bites on 217 of 397 generators — 199 print a residual and 18 are exempt with a published reason; 180 factorise nothing. Read from lib/residual-rule.js, which is the same body the gate enforces from, and the gate's last check fails the build if this page and it disagree about any generator.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Orthogonality, measured

A test with no answer in it

A caller with no reference answer can still ask whether a routine answered the right question: reverse the columns, run it again, compare. The polar factor's two answers agree to 10⁻¹⁵ at every conditioning drawn; a QR's differ by 2.353 on matrices whose own norm is 2.449. The test has a floor, and the floor is measurable too.

 Orthogonality, measured

Two Gram–Schmidts

One argument changes. Classical Gram–Schmidt projects the original column onto each previous direction; modified projects what is left of it. In exact arithmetic the coefficients are identical. In floating point they differ by eight orders of magnitude in the thing that matters.

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