Speaker
Description
This 20-minute contributed talk focuses on two properties introduced in recent work on matrix manifolds: the Row Equivalence Transversality Property (RETP) and the Column Equivalence Transversality Property (CETP).
For an $ m \times n $ real matrix $ A $, RETP holds if the manifolds $ \{GA : G \in \mathrm{GL}(m, \mathbb{R})\} $ and $ Q(\mathrm{sgn}(A)) $ intersect transversally at $ A $, with tangent space $ \{YA : Y \in \mathbb{R}^{m \times m}\} $. Similarly, CETP involves $ \{AG : G \in \mathrm{GL}(n, \mathbb{R})\} $ with tangent space $ \{AX : X \in \mathbb{R}^{n \times n}\} $.
We establish that RETP and CETP each imply RPTP, but the converse does not hold (e.g., certain block matrices have RPTP without RETP/CETP). Key results include:
Theorem 4.1: $ A $ has RETP iff for each row with zeros, the corresponding columns are linearly independent.
Theorem 4.2: Dual for CETP with rows.
Dimension bounds: If RETP, then $ m \cdot \mathrm{rank}(A) + \#(A) \geq mn $; similar for CETP.
Invariance under permutation and diagonal equivalence.
Theorem 4.4: RPTP with zero row/column implies full rank and RETP/CETP.
Corollaries for partitioning matrices to verify RPTP via RETP/CETP (e.g., Theorem 4.8, Corollary 4.9).
Examples: Matrices with RPTP but not RETP/CETP; conditions for both.
These properties aid in studying minimum ranks of sign and zero-nonzero patterns, with connections to transverse intersections in smooth manifolds. No prior RPTP knowledge assumed; proofs and examples provided.