sep(A, B) and the smallest |λᵢ(A) + μⱼ(B)| as the departure from normality grows, n = 6
At its defaults it draws sep(a, b) and the smallest |λᵢ(a) + μⱼ(b)| as the departure from normality grows, n = 6. A and B are upper triangular with diagonals 1, 2, …, 6 and μ above them, so every eigenvalue sum is at least 2 at every point on the axis and the flat line is exact rather than nearly flat. sep — the smallest Frobenius norm of AX + XB over X of unit Frobenius norm, and the reciprocal of the amplification a perturbation of C receives — starts equal to the gap at μ = 0, where the matrices are normal, and falls to 9.47·10⁻⁴ at μ = 8. The number a reader is invited to consult is the one that does not move.
sep-vs-gap is one function in lib/figures/sylvester.js —
matrix equations — the n²×n² coefficient matrix nobody forms, and sep. 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.
A and B are upper triangular with diagonals 1, 2, …, 6 and μ above them, so every eigenvalue sum is at least 2 at every point on the axis and the flat line is exact rather than nearly flat. sep — the smallest Frobenius norm of AX + XB over X of unit Frobenius norm, and the reciprocal of the amplification a perturbation of C receives — starts equal to the gap at μ = 0, where the matrices are normal, and falls to 9.47·10⁻⁴ at μ = 8. The number a reader is invited to consult is the one that does not move.
n: 6
The arguments are the ones An equation whose unknown is a matrix passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A and B are upper triangular with diagonals 1, 2, …, 6 and μ above them, so every eigenvalue sum is at least 2 at every point on the axis and the flat line is exact rather than nearly flat. sep — the smallest Frobenius norm of AX + XB over X of unit Frobenius norm, and the reciprocal of the amplification a perturbation of C receives — starts equal to the gap at μ = 0, where the matrices are normal, and falls to 9.47·10⁻⁴ at μ = 8. The number a reader is invited to consult is the one that does not move.
n: 3
The arguments are the ones An equation whose unknown is a matrix passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A and B are upper triangular with diagonals 1, 2, …, 3 and μ above them, so every eigenvalue sum is at least 2 at every point on the axis and the flat line is exact rather than nearly flat. sep — the smallest Frobenius norm of AX + XB over X of unit Frobenius norm, and the reciprocal of the amplification a perturbation of C receives — starts equal to the gap at μ = 0, where the matrices are normal, and falls to 0.0251 at μ = 8. The number a reader is invited to consult is the one that does not move.
n: 4
The arguments are the ones An equation whose unknown is a matrix passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A and B are upper triangular with diagonals 1, 2, …, 4 and μ above them, so every eigenvalue sum is at least 2 at every point on the axis and the flat line is exact rather than nearly flat. sep — the smallest Frobenius norm of AX + XB over X of unit Frobenius norm, and the reciprocal of the amplification a perturbation of C receives — starts equal to the gap at μ = 0, where the matrices are normal, and falls to 0.00588 at μ = 8. The number a reader is invited to consult is the one that does not move.
n: 5
The arguments are the ones An equation whose unknown is a matrix passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A and B are upper triangular with diagonals 1, 2, …, 5 and μ above them, so every eigenvalue sum is at least 2 at every point on the axis and the flat line is exact rather than nearly flat. sep — the smallest Frobenius norm of AX + XB over X of unit Frobenius norm, and the reciprocal of the amplification a perturbation of C receives — starts equal to the gap at μ = 0, where the matrices are normal, and falls to 0.00203 at μ = 8. The number a reader is invited to consult is the one that does not move.
n: 8
The arguments are the ones An equation whose unknown is a matrix passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A and B are upper triangular with diagonals 1, 2, …, 8 and μ above them, so every eigenvalue sum is at least 2 at every point on the axis and the flat line is exact rather than nearly flat. sep — the smallest Frobenius norm of AX + XB over X of unit Frobenius norm, and the reciprocal of the amplification a perturbation of C receives — starts equal to the gap at μ = 0, where the matrices are normal, and falls to 4.02·10⁻⁴ at μ = 8. The number a reader is invited to consult is the one that does not move.
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.
19 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 smallest |λᵢ + μⱼ| is 2 at μ = 0, unchanged — checked 8 times
and it falls again at μ = 0.5 — checked 7 times
a size the n²×n² singular value decomposition can afford
and away from normality sep falls by orders of magnitude
at μ = 0 the matrices are normal and sep IS the eigenvalue gap
matmul shapes agree
Against the rule
It draws a decomposition and prints its residual. It calls
svd,
and every figure above carries the badge — which residualcheck verifies by looking
for it in the emitted SVG rather than by finding the call that builds one. A badge that is
constructed and then left out of the body is the failure that check exists for.
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.