real-schur-form
At its defaults it draws the real schur form with 2 conjugate pairs: 2 blocks that cannot be split. A square matrix drawn as a grid. Everything below the diagonal is zero except for a small number of two-by-two boxes on the diagonal, which are highlighted.
real-schur-form 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 square matrix drawn as a grid. Everything below the diagonal is zero except for a small number of two-by-two boxes on the diagonal, which are highlighted.
pairs: 1
The arguments are the ones A condition number for one eigenvalue 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 square matrix drawn as a grid. Everything below the diagonal is zero except for a small number of two-by-two boxes on the diagonal, which are highlighted.
pairs: 2
The arguments are the ones The form a real matrix can reach 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 square matrix drawn as a grid. Everything below the diagonal is zero except for a small number of two-by-two boxes on the diagonal, which are highlighted.
pairs: 0
The arguments are the ones The form a real matrix can reach 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 square matrix drawn as a grid. Everything below the diagonal is zero except for a small number of two-by-two boxes on the diagonal, which are highlighted.
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.
8 distinct claims across 4 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 = ZTZᵀ
and every eigenvalue is the one built in
and Z is orthogonal
between zero and three conjugate pairs
matmul shapes agree
one 2×2 block per conjugate pair, and no others
the Francis iteration converged
the grid leaves enough width for an entry
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 52
of 99 generators —
37 print a residual and
15 are exempt with a published reason;
47 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, rankThe form a real matrix can reach
A real matrix with complex eigenvalues has no real triangular form, and the reason is one line — a real triangular matrix has a real diagonal, and a similarity does not move the spectrum. What it has instead is triangular except for one two-by-two block per conjugate pair, and the count is decided by the matrix rather than by where the iteration stopped.
Eigenvalues, singular values, rankTwo shifts that are never formed
The double shift is defined as a factorisation of (A − μI)(A − μ̄I), which nobody computes. What is computed is the first column of that product — three numbers — and the bulge those three numbers create, pushed down the subdiagonal by n − 2 reflectors until it falls off the bottom.