sep-vs-gap
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 ‖AX + XB‖_F over ‖X‖_F = 1, 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 ‖AX + XB‖_F over ‖X‖_F = 1, 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 eigenvalue one vector cannot see passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run 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 ‖AX + XB‖_F over ‖X‖_F = 1, 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: 8
The arguments are the ones An equation whose unknown is a matrix passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run 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 ‖AX + XB‖_F over ‖X‖_F = 1, 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 asserted its own claims on the way to being drawn, and a claim that failed
would have failed the build rather than drawn a wrong picture. Those assertions 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 3 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 — asserted 8 times
and it falls again at μ = 0.5 — asserted 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 92
of 198 generators —
77 print a residual and
15 are exempt with a published reason;
106 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.
An eigenvalue one vector cannot see
A matrix with an exactly doubled eigenvalue at 10. Twelve Lanczos steps find it once; twenty-four find it once, on a Krylov space of dimension 23 in a 24-dimensional problem. A block of two vectors finds it twice. This is not slow convergence — the second copy is not in the space.
Structure, and the solver that cannot see itAn equation whose unknown is a matrix
AX + XB = C is linear in X, so it has a coefficient matrix, and writing it down is the obvious thing to do. At n = 100 that matrix has a hundred million entries for a problem with ten thousand unknowns, and the algorithm everybody uses instead never forms it. Its conditioning is not the eigenvalue gap either, which is the number a reader is invited to consult.
Eigenvalues, singular values, rankHow wide the block should be
A block narrower than the multiplicity does not converge slowly — it never returns the missing copy at all. Above the multiplicity every extra column buys iterations at about ten products with A each. And the mechanism that is supposed to make the choice unimportant never fires from a random start.
Eigenvalues, singular values, rankThe eigenvalues that are not there
For a normal matrix the resolvent norm is exactly one over the distance to the nearest eigenvalue, so a picture of it carries nothing the spectrum did not. Move one entry above the diagonal and the region a perturbation of 10⁻⁸ can put an eigenvalue into stops being a disc and reaches out past the unit circle, while every eigenvalue stays at 0.8.
Eigenvalues, singular values, rankThe gap decides the eigenvector
A symmetric matrix's eigenvalues move by at most the size of the perturbation, whatever the spectrum looks like. Its eigenvectors are governed by a completely different quantity — the distance to the neighbouring eigenvalue — and at a gap of 10⁻⁹ the same perturbation turns them through 27°.
Eigenvalues, singular values, rankThe plane survives what its vectors do not
At a gap of 10⁻⁹ a perturbation of 10⁻⁶ turns the two eigenvectors through half a radian and turns the plane they span through 7.6·10⁻⁸ — a ratio of six million. Ask for the subspace instead of the vectors and a hopeless computation becomes a well-conditioned one, with no change to the arithmetic.