4 Invariance
\(p_k(\ell \cdot J) = \det (\ell _k) \cdot p_k(J)\) for lower-triangular \(\ell \).
Proof
Block structure of lower-triangular matrices.
\(q_{k,l}(\ell ' \cdot J) = \det (\ell '_k) \cdot q_{k,l}(J)\).
Proof
Similar to the leading principal minor case.
\(\Pi \)-stability is invariant under left-multiplication by a positive diagonal matrix.
Proof
The \(r\)-minor scales by a positive factor.
\(\Pi \)-compatibility is invariant under left-multiplication by a positive diagonal matrix.
Proof
The \(q\)-minor scales by a positive factor.