1 Minors and matrix properties
The \(k\)-th leading principal minor \(p_k(J)\) is the determinant of the leading principal \(k \times k\) submatrix of \(J\).
The minor \(q_{k,l}(J)\) is the determinant of the submatrix with rows \(\{ 0,\ldots ,k{-}1\} \) and columns \(\{ 0,\ldots ,k{-}2, l\} \).
The minor \(r_{j,k}(J)\) is the determinant of the \((k{-}1)\times (k{-}1)\) submatrix with rows \(\{ 0,\ldots ,k{-}1\} \setminus \{ j\} \) and columns \(\{ 0,\ldots ,k{-}2\} \).
A matrix \(J\) is stable if all leading principal minors \(p_1,\ldots ,p_n\) are positive and the signed \(r\)-minors \((-1)^{k-j} r_{j,k}\) are non-negative.
A matrix \(J\) is compatible if it is either not stable, or for all \(k\) and \(l {\gt} k\) we have \(q_{k,l}(J) \leq 0\).
Let \(J = LU\) with \(L\) unit lower triangular and \(U\) upper triangular, and suppose \(J\) is stable. Then \(J\) is compatible if and only if every strictly-above-diagonal entry of \(U\) is non-positive. This rests on the identity \(q_{k,l}(LU) = \bigl(\prod _{j{\lt}k-1} u_{jj}\bigr)\, u_{k-1,l}\).
Factor \(L\) out of the \(q\)-minor submatrix; the remaining \(U\)-block is upper triangular, so its determinant is the product of its diagonal entries. Under stability the diagonal of \(U\) is positive, so the sign of \(q_{k,l}\) is the sign of \(u_{k-1,l}\).
A matrix \(J\) is soluble if all leading principal minors \(p_1,\ldots ,p_n\) are nonzero.
\(p_{k+1}(J) = J_{0,0} \cdot p_k(\mathrm{Schur}_{00}(J))\).
Block matrix factorization via Mathlib’s Schur complement.
\(r_{j+1,k+2}(J) = J_{0,0} \cdot r_{j,k+1}(\mathrm{Schur}_{00}(J))\).
Uses Laplace vanishing and the Schur complement structure.
The imaginary numerator satisfies \(\mathrm{imNum}(J, k{+}1, t) = J_{0,0} \cdot \mathrm{imNum}(\mathrm{Schur}_{00}(J), k, t')\) where \(t'\) is the Schur-restricted parameter.
Combines the leading principal minor and \(r\)-minor Schur identities.
\(J\) is stable if and only if it is stable at every step.
Direct unfolding of definitions.
If \(J\) is stable, then \(J\) is soluble.
Positive minors are nonzero.