directed cycle on 18: a Laplacian whose eigenvalues need a plane

At its defaults it draws directed cycle on 18: a laplacian whose eigenvalues need a plane. The spectrum of L = D_out − A for a directed cycle on 18, drawn in the complex plane. The row sums are exactly zero — measured at 0, not at the rounding level — so 1 is a right null vector and 0 is an eigenvalue, computed here at 7.53·10⁻¹⁶. The column sums are not: the worst is 0, and the left null vector is the stationary distribution rather than 1. The largest imaginary part is 0.9848 and the matrix is 100 per cent asymmetric in the Frobenius norm — and those two numbers are not the same statement: asymmetry permits a complex spectrum and does not force one, which this family demonstrates at small sizes by being asymmetric and real. The ring is the closed form 1 − exp(2πik/n), which every computed eigenvalue lands on to 1.3·10⁻¹⁵.

directed-spectrum is one function in lib/figures/digraph.js — arrows — a laplacian that is not symmetric, and the walk that has to be computed before it can be. 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.

The spectrum of L = D_out − A for a directed cycle on 18, drawn in the complex plane. The row sums are exactly zero — measured at 0, not at the rounding level — so 1 is a right null vector and 0 is an eigenvalue, computed here at 7.53·10⁻¹⁶. The column sums are not: the worst is 0, and the left null vector is the stationary distribution rather than 1. The largest imaginary part is 0.9848 and the matrix is 100 per cent asymmetric in the Frobenius norm — and those two numbers are not the same statement: asymmetry permits a complex spectrum and does not force one, which this family demonstrates at small sizes by being asymmetric and real. The ring is the closed form 1 − exp(2πik/n), which every computed eigenvalue lands on to 1.3·10⁻¹⁵.

The spectrum of L = D_out − A for a two blocks with 1 arc back, drawn in the complex plane. The row sums are exactly zero — measured at 0, not at the rounding level — so 1 is a right null vector and 0 is an eigenvalue, computed here at 1.03·10⁻¹⁵. The column sums are not: the worst is 2, and the left null vector is the stationary distribution rather than 1. The largest imaginary part is 0.1864 and the matrix is 24 per cent asymmetric in the Frobenius norm — and those two numbers are not the same statement: asymmetry permits a complex spectrum and does not force one, which this family demonstrates at small sizes by being asymmetric and real. There is no closed form for this family; the degenerate check is the symmetric one, where every imaginary part is exactly zero.

The spectrum of L = D_out − A for a random strongly connected on 30, drawn in the complex plane. The row sums are exactly zero — measured at 0, not at the rounding level — so 1 is a right null vector and 0 is an eigenvalue, computed here at 5.98·10⁻¹⁶. The column sums are not: the worst is 3, and the left null vector is the stationary distribution rather than 1. The largest imaginary part is 1.174 and the matrix is 70 per cent asymmetric in the Frobenius norm — and those two numbers are not the same statement: asymmetry permits a complex spectrum and does not force one, which this family demonstrates at small sizes by being asymmetric and real. There is no closed form for this family; the degenerate check is the symmetric one, where every imaginary part is exactly zero.

The spectrum of L = D_out − A for a balanced digraph on 24, drawn in the complex plane. The row sums are exactly zero — measured at 0, not at the rounding level — so 1 is a right null vector and 0 is an eigenvalue, computed here at 3.89·10⁻¹⁶. The column sums are not: the worst is 0, and the left null vector is the stationary distribution rather than 1. The largest imaginary part is 1.505 and the matrix is 82.5 per cent asymmetric in the Frobenius norm — and those two numbers are not the same statement: asymmetry permits a complex spectrum and does not force one, which this family demonstrates at small sizes by being asymmetric and real. There is no closed form for this family; the degenerate check is the symmetric one, where every imaginary part is exactly zero.

The spectrum of L = D_out − A for a directed cycle on 24, drawn in the complex plane. The row sums are exactly zero — measured at 0, not at the rounding level — so 1 is a right null vector and 0 is an eigenvalue, computed here at 2.37·10⁻¹⁶. The column sums are not: the worst is 0, and the left null vector is the stationary distribution rather than 1. The largest imaginary part is 1 and the matrix is 100 per cent asymmetric in the Frobenius norm — and those two numbers are not the same statement: asymmetry permits a complex spectrum and does not force one, which this family demonstrates at small sizes by being asymmetric and real. The ring is the closed form 1 − exp(2πik/n), which every computed eigenvalue lands on to 1.3·10⁻¹⁵.

The spectrum of L = D_out − A for a directed cycle on 18, drawn in the complex plane. The row sums are exactly zero — measured at 0, not at the rounding level — so 1 is a right null vector and 0 is an eigenvalue, computed here at 7.53·10⁻¹⁶. The column sums are not: the worst is 0, and the left null vector is the stationary distribution rather than 1. The largest imaginary part is 0.9848 and the matrix is 100 per cent asymmetric in the Frobenius norm — and those two numbers are not the same statement: asymmetry permits a complex spectrum and does not force one, which this family demonstrates at small sizes by being asymmetric and real. The ring is the closed form 1 − exp(2πik/n), which every computed eigenvalue lands on to 1.3·10⁻¹⁵.

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.

13 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 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 214 of 397 generators — 194 print a residual and 20 are exempt with a published reason; 183 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.