kernel_dimension

std.linalg.kernel_dimension · Level L4

The rank of an integer matrix A (m × n, a linear map Rⁿ → Rᵐ) and the dimension of its kernel, n − rank by rank-nullity, decided exactly. The kernel is the solution set of Ax = 0; a nonzero right-hand side gives an affine set, not the kernel.

dim ker A = n − rank A

Signature

kernel_dimension(A: f64[m, n]) → f64[], f64[]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference exact rank, and n minus it in exact rational arithmetic, on all 40 test cases.
All 80 float64 results are exact: the error is zero.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them exact, with zero error.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 100%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 0

Identity

Calls

rank_exact

Called by

—

sha256:d17255dadb29582d435bf4cc8673c3db0c106e627a635392d079184afee28a8e

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.linalg.kernel_dimension · Ω:kernel_dimension
Pins: sha256:56142e9ef1d3d5d9b7be9442af34b33064eb38cb2437746bc1b8bda419d282d6this graph and the 2 it reaches through calls
Evidence: sha256:1b768d955b032c39ade55362303c7721f30f7471a4b5b83d7de5ec367acee685the hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.