Core EP, Dual Core EP and Composite Generalized Inverses for a Class of Structured Matrices

1. Problem Statement

The paper addresses the problem of characterizing and explicitly computing various types of generalized inverses for matrices associated with double star digraphs.

• Context: Generalized inverses (such as Moore–Penrose, Drazin, and Group inverses) are essential for solving singular systems, constrained least squares problems, and spectral decompositions. While standard inverses exist for non-singular matrices, many structured matrices arising in network theory and combinatorics are singular or have high indices.
• Specific Object: The authors focus on matrices $M$ derived from double star digraphs $S_{(m+1),(n+1)}$. These graphs consist of two central vertices ($u$ and $v$) connected by bidirectional edges, where each center is also connected to a set of pendant vertices (leaves).
• Gap: While formulas for standard inverses (like the Group inverse) for these matrices exist, there is a lack of explicit closed-form expressions and existence criteria for more advanced core-type inverses (Core, Dual Core, Core EP, Dual Core EP) and their composite versions (MPCEP, $\ast$CEPMP, GDC, GC) specifically for this structured class, particularly in cases where the Group inverse does not exist.

2. Methodology

The authors employ a combination of block matrix algebra, Schur complement techniques, and full rank factorizations to derive their results.

• Canonical Representation: They establish that any matrix $M$ associated with a double star digraph is permutation similar (and thus unitarily similar) to a specific canonical block form (Equation 2.1). This matrix is partitioned into blocks involving vectors $x, y, z, w$ and scalars $a, b$.
  $$M = \begin{bmatrix} 0 & x^T & a & 0 \\ y & 0 & 0 & 0 \\ 0 & 0 & 0 & z^T \\ 0 & 0 & w & 0 \end{bmatrix}$$
  Note: The actual block structure in the paper is slightly more complex, separating the central vertices from the leaves, but the logic relies on this block decomposition.

• Parameterization: The analysis relies on defining specific scalar parameters derived from the vectors and scalars in the matrix:

  • $s = x^*x, \quad t = z^*z, \quad u = y^*y, \quad v = w^*w$ (Gram scalars).
  • $\zeta = x^T y + ab$ (Coupling term).
  • Derived scalars like $r, h, p, q, \alpha, \beta$ involving combinations of the above.

• Case Analysis: The study is divided based on the Drazin index $i(M)$ and the existence of the Group inverse ($M^\#$). The authors identify three mutually exclusive cases:

  1. Case I: $x^T y = 0$ and $z^T w = 0$ (Index $i(M)=2$).
  2. Case II: $x^T y \neq 0, z^T w = 0$ (or vice versa), with $\zeta \neq 0$ (Index $i(M)=3$).
  3. Case III: $x^T y \neq 0, z^T w = 0$, with $\zeta = 0$ (Index $i(M)=5$, Nilpotent).

• Derivation Strategy:

  • For Core/Dual Core inverses: They utilize the condition that these exist if and only if both the Moore–Penrose inverse ($M^\dagger$) and the Group inverse ($M^\#$) exist.
  • For Core EP/Dual Core EP inverses: They use Lemma 2.2, which relates these inverses to the Drazin inverse ($M^D$) and the existence of $\{1,3\}$-inverses (for Core EP) or $\{1,4\}$-inverses (for Dual Core EP) of powers of $M$ (specifically $M^k$ where $k \geq i(M)$).
  • Explicit Construction: They compute the necessary powers ($M^2, M^3$), analyze the rank of Gram-type products ($M^k(M^k)^*$ and $(M^k)^*M^k$) to determine existence, and construct explicit von Neumann inverses via full rank factorizations to derive the final block formulas.

3. Key Contributions

The paper makes several significant theoretical and computational contributions:

1. Explicit Block Formulas: The authors provide closed-form block matrix expressions for:

   • Core Inverse ($M^{\# \circ}$) and Dual Core Inverse ($M^{\circ \#}$).
   • Core EP Inverse ($M^{\text{d}\circ}$) and Dual Core EP Inverse ($M^{\text{d} \circ}$).
   • Composite inverses: MPCEP ($M^{\dagger, \text{d}\circ}$), $\ast$CEPMP ($M^{\text{d} \circ, \dagger}$), Generalized Dual Core (GDC), and Generalized Core (GC).

2. Algebraic Existence Criteria: They derive precise algebraic conditions (in terms of the non-vanishing of the scalars $s, u, t, v, r, h, p, q, \alpha, \beta$) required for each inverse to exist. This clarifies exactly when these generalized inverses are well-defined for double star matrices.

3. Handling High-Index Matrices: A major contribution is the treatment of matrices where the Group inverse does not exist (Cases I and II). In these scenarios, the Core EP and Dual Core EP inverses serve as the appropriate generalizations, and the paper provides the first explicit formulas for them in this context.

4. Structural Insights: The paper demonstrates that for double star digraphs, the existence of these inverses is strictly tied to the non-singularity of specific Gram scalars and coupling terms. It also proves that for this specific class of matrices, the composite inverses (e.g., MPCEP vs. $\ast$CEPMP) never coincide unless the matrix is trivial (zero vectors), highlighting the distinct nature of these operators.

4. Key Results

• Case I (Index 2):

  • $M^{\text{d}\circ}$ exists iff $r \neq 0$ and $h \neq 0$.
  • $M^{\text{d} \circ}$ exists iff $p \neq 0$ and $q \neq 0$.
  • Explicit formulas for MPCEP, $\ast$CEPMP, GDC, and GC are derived under the combined conditions of Moore–Penrose and Core EP/Dual Core EP existence.

• Case II (Index 3):

  • $M^{\text{d}\circ}$ exists iff $h \neq 0$ and $\beta \neq 0$.
  • $M^{\text{d} \circ}$ exists iff $q \neq 0$ and $\alpha \neq 0$.
  • Similar explicit formulas are provided for the composite inverses, showing dependence on the coupling term $\zeta$.

• Case III (Index 5):

  • The matrix is nilpotent ($M^5=0$).
  • The Drazin inverse is zero.
  • Consequently, all Core EP, Dual Core EP, and composite inverses are the zero matrix.

• Summary Table: The paper concludes with a comprehensive table (Table 1) summarizing the existence conditions for all studied inverses across the non-trivial cases, serving as a quick reference for researchers.

5. Significance

• Theoretical Advancement: This work extends the algebraic theory of generalized inverses from standard rings and simple matrices to complex, structured matrices arising from graph theory. It bridges the gap between combinatorial graph structures and advanced linear algebraic operators.
• Computational Utility: By providing explicit block formulas, the paper allows for the direct computation of these inverses without resorting to iterative numerical methods, which is crucial for large-scale network analysis or symbolic computation.
• Classification of Singular Dynamics: The detailed case analysis helps in understanding the behavior of singular systems modeled by double star digraphs. Knowing the exact conditions for the existence of Core EP inverses allows researchers to determine if a system admits a specific type of spectral decomposition or projection.
• Foundation for Future Work: The methodology of using Schur complements and full rank factorizations on block-structured matrices can be adapted to other graph families (e.g., multi-star graphs, trees with specific coupling), potentially leading to a broader theory of generalized inverses for structured networks.

In summary, the paper provides a rigorous, complete, and explicit characterization of a wide spectrum of generalized inverses for double star digraph matrices, resolving existence questions and providing ready-to-use formulas for both theoretical analysis and practical computation.