The condition number of one eigenvalue, at an off-diagonal entry of 100
At its defaults it draws the condition number of one eigenvalue, at an off-diagonal entry of 100. A rising curve against the size of an off-diagonal entry, on logarithmic axes, with a flat line at one below it and a vertical marker at the current value.
eigenvalue-conditioning is one function in lib/figures/francis.js —
francis — the real schur form and the two shifts that are never formed. 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 rising curve against the size of an off-diagonal entry, on logarithmic axes, with a flat line at one below it and a vertical marker at the current value.
c: 100
The arguments are the ones A condition number for one eigenvalue passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A rising curve against the size of an off-diagonal entry, on logarithmic axes, with a flat line at one below it and a vertical marker at the current value.
c: 5
The arguments are the ones A condition number for one eigenvalue passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A rising curve against the size of an off-diagonal entry, on logarithmic axes, with a flat line at one below it and a vertical marker at the current value.
c: 1
The arguments are the ones A condition number for one eigenvalue passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A rising curve against the size of an off-diagonal entry, on logarithmic axes, with a flat line at one below it and a vertical marker at the current value.
c: 10
The arguments are the ones A condition number for one eigenvalue passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A rising curve against the size of an off-diagonal entry, on logarithmic axes, with a flat line at one below it and a vertical marker at the current value.
c: 1000
The arguments are the ones A condition number for one eigenvalue passes. A value nobody placed would be a picture no essay asked for and no claim was ever checked against.
A rising curve against the size of an off-diagonal entry, on logarithmic axes, with a flat line at one below it and a vertical marker at the current value.
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.
33 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 condition number at c = 1.0 agree — checked 25 times
A = ZTZᵀ
an off-diagonal entry between one and a thousand
and the departure from normality is the entry itself
and the worst of twelve directions reaches a good fraction of it
LU is for square matrices
matmul shapes agree
no direction moves it by more than the condition number permits
the decoupled eigenvalues are perfectly conditioned throughout
Against the rule
It draws a decomposition and prints its residual. It calls
realSchur,
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.
A condition number for one eigenvalue
In the symmetric case every eigenvalue has condition number exactly one. In this four-by-four matrix two of them have condition number 100.005 and the other two have exactly 1, and the number belongs to the eigenvalue rather than to the matrix.
Eigenvalues, singular values, rankA 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.