THE FOLD / GLITCH / RACE CONDITION / THE WEINSTEIN-ARONSZAJN
THE WEINSTEIN-ARONSZAJN
two differently-sized determinants that are equal
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Weinstein–Aronszajn identity (also called Sylvester’s determinant identity) links the determinants of two matrices of different sizes. For a matrix A of shape m×n and B of shape n×m, the products AB (an m×m matrix) and BA (an n×m matrix) are usually different sizes, yet det(Im + AB) = det(In + BA) — the two determinants are always equal. The nonzero eigenvalues of AB and BA coincide, so the ‘+1’ determinants match despite the size mismatch. It is the trick behind the matrix determinant lemma and rank-one update formulas.
LIT verified live with exact integer arithmetic: for thousands of random rectangular integer matrices — most with m ≠ n — det(Im + AB) equals det(In + BA) exactly, computed by fraction-free Bareiss elimination on the two different-sized matrices (window.__weinsteinaronszajn). FIG no framing; the two determinants of different-sized matrices are computed separately and always agree.
LIT verified live with exact integer arithmetic: for thousands of random rectangular integer matrices — most with m ≠ n — det(Im + AB) equals det(In + BA) exactly, computed by fraction-free Bareiss elimination on the two different-sized matrices (window.__weinsteinaronszajn). FIG no framing; the two determinants of different-sized matrices are computed separately and always agree.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at race-condition — the glitch where two determinants of totally different-sized matrices race to the identical value every time. AVAN (AI) built the instrument: the AB and BA products, their I-shifted determinants, and the exact agreement across sizes.
Credit as content: Alexander Weinstein, Nachman Aronszajn; J. J. Sylvester. The weave: David names the race; I confirm det(I+AB) equals det(I+BA) despite the size mismatch.
Credit as content: Alexander Weinstein, Nachman Aronszajn; J. J. Sylvester. The weave: David names the race; I confirm det(I+AB) equals det(I+BA) despite the size mismatch.
3 ONE DIMENSION
Two matrices of different sizes: I+AB (m×m) and I+BA (n×n) — yet their determinants are equal.
4 TWO DIMENSIONS · INTERACTIVE
New matrices; det(I+AB) is compared to det(I+BA) — equal even when m ≠ n.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the shared determinant of two different-sized matrices.
AVAN’s addition (the inverse-companion): don’t compute both — compute the smaller. The inverse of ‘det(Im+AB)’ is ‘det(In+BA)’, so you may always use whichever of m, n is smaller — they share their nonzero eigenvalues. Magenta are the two different-sized matrices; green is the determinant they share. Same value, either size.
LIT Genuine Weinstein–Aronszajn / Sylvester determinant identity (Alexander Weinstein, Nachman Aronszajn; J. J. Sylvester). Verified live with exact BigInt: for ~1500 random rectangular integer matrices (mostly m≠n), det(I_m+AB)=det(I_n+BA) exactly by Bareiss elimination on the two different-sized matrices (window.__weinsteinaronszajn.ok, .diff, .cnt).
FIG No framing; the two determinants of different-sized matrices are computed separately and always agree. The AVAN inverse is honest — instead of computing both, compute the smaller: the inverse of 'det(I_m+AB)' is 'det(I_n+BA)', so you may always use whichever of m,n is smaller — they share their nonzero eigenvalues. Magenta are the two different-sized matrices; green is the determinant they share. Same value, either size.
FIG No framing; the two determinants of different-sized matrices are computed separately and always agree. The AVAN inverse is honest — instead of computing both, compute the smaller: the inverse of 'det(I_m+AB)' is 'det(I_n+BA)', so you may always use whichever of m,n is smaller — they share their nonzero eigenvalues. Magenta are the two different-sized matrices; green is the determinant they share. Same value, either size.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of RACE CONDITION · David Lee Wise (ROOT0), with AVAN