One 6 × 6 integer matrix, 3 different ranks
At its defaults it draws one 6 × 6 integer matrix, 3 different ranks. The same matrix of integers, its rank computed exactly with no rounding and no threshold anywhere, over the rationals and over the fields with 2, 3, 5, 7, 11, 13, 101, 65537 elements. Over ℚ the rank is 6; over 𝔽2 and 𝔽3 it is 4 and 5. Nothing has been approximated and nothing has been decided by a tolerance — the eliminations are exact in every case, and there is no small number anywhere for a threshold to be compared against. The rank is not a property of the array of numbers; it is a property of the array together with the ring the entries are read in, and a modular algorithm that reports a rank has reported the second one.
rank-by-field is one function in lib/figures/integer.js —
exact arithmetic — no residual to print, and a cost measured in the length of the numbers. 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.
The same matrix of integers, its rank computed exactly with no rounding and no threshold anywhere, over the rationals and over the fields with 2, 3, 5, 7, 11, 13, 101, 65537 elements. Over ℚ the rank is 6; over 𝔽2 and 𝔽3 it is 4 and 5. Nothing has been approximated and nothing has been decided by a tolerance — the eliminations are exact in every case, and there is no small number anywhere for a threshold to be compared against. The rank is not a property of the array of numbers; it is a property of the array together with the ring the entries are read in, and a modular algorithm that reports a rank has reported the second one.
n: 6
The arguments are the ones A prime that divides the answer passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
The same matrix of integers, its rank computed exactly with no rounding and no threshold anywhere, over the rationals and over the fields with 2, 3, 5, 7, 11, 13, 101, 65537 elements. Over ℚ the rank is 6; over 𝔽2 and 𝔽3 it is 4 and 5. Nothing has been approximated and nothing has been decided by a tolerance — the eliminations are exact in every case, and there is no small number anywhere for a threshold to be compared against. The rank is not a property of the array of numbers; it is a property of the array together with the ring the entries are read in, and a modular algorithm that reports a rank has reported the second one.
show: "field-shares"
The arguments are the ones The field decides it, usually passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
3000 seeded n by n matrices at each size, each entry 1 with probability a half. over the rationals: 0.38, 0.34, 0.35, 0.42, 0.55, 0.71, 0.84, 0.93, 0.97, 1.00; over the field of two elements: 0.38, 0.33, 0.31, 0.29, 0.29, 0.29, 0.30, 0.29, 0.29, 0.29; over the field of three elements: 0.38, 0.34, 0.35, 0.39, 0.45, 0.52, 0.55, 0.55, 0.55, 0.56; over the field of five elements: 0.38, 0.34, 0.35, 0.42, 0.53, 0.65, 0.72, 0.74, 0.75, 0.74; over the field of seven elements: 0.38, 0.34, 0.35, 0.42, 0.54, 0.69, 0.77, 0.81, 0.83, 0.83 at n = 2, 3, 4, 6, 8, 10, 12, 14, 16, 20. The dashed lines are the invertible share of a matrix uniform over each field; at n = 20 they are 0.289, 0.560, 0.760, 0.837.
show: "field-lost"
The arguments are the ones The field decides it, usually passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A matrix invertible over the rationals is singular modulo p exactly when p divides its determinant. Modulo two: 0.03, 0.10, 0.31, 0.47, 0.59, 0.65, 0.68, 0.70, 0.71; Modulo three: 0.00, 0.01, 0.06, 0.17, 0.27, 0.35, 0.41, 0.43, 0.43; Modulo five: 0.00, 0.00, 0.00, 0.02, 0.09, 0.15, 0.20, 0.22, 0.26; Modulo seven: 0.00, 0.00, 0.00, 0.01, 0.04, 0.09, 0.12, 0.14, 0.17 at n = 3, 4, 6, 8, 10, 12, 14, 16, 20. The dashed lines are one minus the uniform law's invertible share, which at n = 20 is 0.711, 0.440, 0.240, 0.163.
show: "field-corank", p: 3
The arguments are the ones The field decides it, usually passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
The share of 3000 matrices whose rank modulo 3 falls short of 20 by 0, 1, 2 and 3: 0.5650, 0.4170, 0.0180, 0.0000. The law for large uniform matrices over the field: 0.5601, 0.4201, 0.0197, 0.0001.
show: "field-density"
The arguments are the ones The field decides it, usually passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
Each entry 1 with probability a quarter, a half or three quarters. Density 0.25: 0.061, 0.076, 0.121, 0.209, 0.237; Density 0.5: 0.313, 0.288, 0.298, 0.288, 0.290; Density 0.75: 0.242, 0.188, 0.216, 0.265, 0.289 at n = 4, 8, 12, 16, 20; the uniform law is 0.308, 0.290, 0.289, 0.289, 0.289.
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.
14 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.
a prime the dial draws
a size between 2 and 24
a size to compute the rank of
an entry density strictly between 0 and 1
and the rank modulo 3 loses the one divisible by three
every Bareiss division is exact
the assertion refuses a counterexample
the example's determinant is even and a multiple of three
the left transform is unimodular
the matrix is nonsingular over the rationals
the rank modulo 2 loses both invariants divisible by two
the rank over a finite field is never larger than the rank over ℚ
the right transform is unimodular
while every prime coprime to the invariants returns the rank over ℚ
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.
A prime that divides the answer
A modular elimination reports a singular matrix and is telling the truth — over the field with p elements the matrix is singular. Over the rationals it is not. Nothing in the residue distinguishes the two cases, no quantity is small enough to be suspicious, and the wrong answer is a correct computation of a different question.
Exact arithmetic, and what it costs insteadThe field decides it, usually
A matrix whose rank depends on the field it is read over was built, the first time, from its invariant factors outward, because random integer matrices never seemed to show the effect. Random 0/1 matrices show it at almost every size that is not tiny. At twenty rows, 99.8% of them are invertible over the rationals, 29% modulo two, 56% modulo three — and 71% of the ones the rationals call invertible are singular modulo two. Modulo two they obey, corank by corank, the law for uniformly random matrices over that field; modulo three and five, which their entries cannot fill, they converge to that field's law anyway.
Exact arithmetic, and what it costs insteadThe rank depends on the ring
A floating-point rank is a decision about a threshold. Remove the arithmetic error entirely and the threshold goes away — and the answer still is not a property of the array of numbers, because one integer matrix has rank six over the rationals, five modulo three and four modulo two, with nothing rounded and nothing decided.