Three perturbations of 10⁻¹⁴, three eigenvectors, one plane
At its defaults it draws three perturbations of 10⁻¹⁴, three eigenvectors, one plane. A circle with three radii at widely different angles, one for each perturbation, and a table of their residuals showing all three satisfy the eigenvalue equation.
undefined-eigenvector is one function in lib/figures/subspace.js —
subspaces — the gap, the eigenvector, and the plane that survives both. 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 circle with three radii at widely different angles, one for each perturbation, and a table of their residuals showing all three satisfy the eigenvalue equation.
size: 1e-14
The arguments are the ones A function of a matrix is not a function of its entries passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A circle with three radii at widely different angles, one for each perturbation, and a table of their residuals showing all three satisfy the eigenvalue equation.
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.
7 distinct claims across 2 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 perturbation far below anything a tolerance would notice
and exactly the same plane
every one of them satisfies Ax = λx
Jacobi needs a symmetric matrix
matmul shapes agree
the two eigenvalues are equal to rounding
three perturbations of the same size return unrelated eigenvectors
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 function of a matrix is not a function of its entries
Everybody learns that f(A) means diagonalise, apply f to the eigenvalues, undiagonalise. That is a definition, not a method. On a matrix seven picometres from a defective one — with exact eigenvalues and eigenvectors from a closed form — the definition returns an answer wrong by sixty-five orders of magnitude, and a method that never mentions an eigenvalue returns the right one.
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°.