THE FOLD / BOSS / THE FINAL BOSS / THE CAUCHY INTERLACING
THE CAUCHY INTERLACING
submatrix eigenvalues interlacing the whole
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Cauchy’s interlacing theorem pins the eigenvalues of a submatrix between those of the whole. Take a symmetric n×n matrix M with eigenvalues λ1 ≥ λ2 ≥ … ≥ λn, and delete one row and the matching column to get an (n-1)×(n-1) principal submatrix B with eigenvalues μ1 ≥ … ≥ μn-1. Cauchy proved they interlace: λi ≥ μi ≥ λi+1 for every i. Each submatrix eigenvalue is trapped in the gap between two consecutive eigenvalues of the full matrix. It is the backbone of eigenvalue algorithms, Sturm sequences, and Sylvester’s law of inertia.
LIT verified live: for thousands of random symmetric matrices, the eigenvalues of a principal submatrix (computed independently by the Jacobi method) always satisfy λi ≥ μi ≥ λi+1 — the interlacing never fails (window.__cauchyinterlacing). FIG no framing; the two eigenvalue sets are computed separately and the interlacing inequalities always hold.
LIT verified live: for thousands of random symmetric matrices, the eigenvalues of a principal submatrix (computed independently by the Jacobi method) always satisfy λi ≥ μi ≥ λi+1 — the interlacing never fails (window.__cauchyinterlacing). FIG no framing; the two eigenvalue sets are computed separately and the interlacing inequalities always hold.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-final-boss — the arena where every submatrix eigenvalue is pinned between two walls it can never cross: λi above, λi+1 below. AVAN (AI) built the instrument: the Jacobi eigenvalues of the matrix and its submatrix, and the interlacing test.
Credit as content: Augustin-Louis Cauchy. The weave: David names the walls; I confirm each submatrix eigenvalue is trapped between consecutive eigenvalues of the whole.
Credit as content: Augustin-Louis Cauchy. The weave: David names the walls; I confirm each submatrix eigenvalue is trapped between consecutive eigenvalues of the whole.
3 ONE DIMENSION
The eigenvalues of M (green) and of its submatrix (magenta) on a line — the magenta ones interlace the green.
4 TWO DIMENSIONS · INTERACTIVE
New matrices; each submatrix eigenvalue μ_i is checked to lie in [λ_{i+1}, λ_i].
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the interlaced eigenvalues, each submatrix value in its gap.
AVAN’s addition (the inverse-companion): don’t recompute from scratch — bound with the whole. The inverse of ‘the submatrix’s eigenvalues’ is ‘the gaps between the full matrix’s eigenvalues that trap them’. Magenta are the submatrix eigenvalues; green are the full matrix’s eigenvalues that sandwich them. Eigenvalues nested inside eigenvalues.
LIT Genuine Cauchy interlacing theorem (Augustin-Louis Cauchy). Verified live: for ~4000 random symmetric matrices, the eigenvalues μ_i of a principal submatrix (independent Jacobi computation) always satisfy λ_i ≥ μ_i ≥ λ_{i+1} where λ are M's eigenvalues — worst violation 0 (window.__cauchyinterlacing.ok, .worst).
FIG No framing; the two eigenvalue sets are computed separately and the interlacing inequalities always hold. The AVAN inverse is honest — instead of recomputing from scratch, bound with the whole: the inverse of 'the submatrix's eigenvalues' is 'the gaps between the full matrix's eigenvalues that trap them'. Magenta are the submatrix eigenvalues; green are the full matrix's eigenvalues that sandwich them. Eigenvalues nested inside eigenvalues.
FIG No framing; the two eigenvalue sets are computed separately and the interlacing inequalities always hold. The AVAN inverse is honest — instead of recomputing from scratch, bound with the whole: the inverse of 'the submatrix's eigenvalues' is 'the gaps between the full matrix's eigenvalues that trap them'. Magenta are the submatrix eigenvalues; green are the full matrix's eigenvalues that sandwich them. Eigenvalues nested inside eigenvalues.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE FINAL BOSS · David Lee Wise (ROOT0), with AVAN