On the Drazin Index of an Anti-Triangular Block Matrix

1. Problem Statement

The paper addresses the challenge of determining the Drazin index (denoted $i(M)$) and the Drazin inverse ($M^D$) for anti-triangular block matrices of the form:
$$M = \begin{bmatrix} A & B \\ C & 0 \end{bmatrix}$$
where $A, B, C$ are square or conformal matrices over a field.

While the Drazin index and inverse for standard block triangular matrices (e.g., $\begin{bmatrix} A & B \\ 0 & D \end{bmatrix}$) are well-established with sharp bounds (e.g., $\max\{i(A), i(D)\} \leq i(M) \leq i(A) + i(D)$), the theory for anti-triangular structures is less developed. Specifically, existing literature often lacks explicit closed-form representations for the Drazin inverse of $M$ and tight bounds for its index under general algebraic constraints on the blocks.

2. Methodology

The authors employ a combination of algebraic decomposition techniques and properties of generalized inverses:

• Additive Decomposition: The matrix $M$ is factored into $M = SR$, where $S = \begin{bmatrix} A & I \\ C & 0 \end{bmatrix}$ and $R = \begin{bmatrix} I & 0 \\ 0 & B \end{bmatrix}$. This allows the application of Cline's Lemma, which relates the Drazin inverse of a product to the Drazin inverse of the reverse product: $(SR)^D = S((RS)^D)^2 R$.
• Auxiliary Matrix Construction: The authors define an auxiliary matrix $\Gamma = RS = \begin{bmatrix} A & I \\ BC & 0 \end{bmatrix}$. They relate $i(M)$ to $i(\Gamma)$ using the inequality $|i(M) - i(\Gamma)| \leq 1$.
• Von Neumann Inverses and Core-Nilpotent Decomposition: The study utilizes von Neumann inverses ($A^-$ satisfying $AA^-A=A$) to construct specific sums of the form $A^2A^- + I - AA^-$. The authors prove that the Drazin index of these sums is invariant under the choice of the von Neumann inverse.
• Orthogonal Sums: The analysis often involves decomposing matrices into orthogonal sums ($X+Y$ where $XY=YX=0$), allowing the index of the sum to be determined as $\max\{i(X), i(Y)\}$.
• Minimal Polynomials: The paper establishes relationships between the minimal polynomials of $M$, its blocks, and the constructed auxiliary matrices to derive index bounds.

3. Key Contributions and Results

A. Invariance Properties of Special Sums

The paper establishes that for a singular matrix $A$ and any von Neumann inverse $A^-$, the quantity $i(A^2A^- + I - AA^-)$ is invariant and equals $i(A) - 1$. This provides a mechanism to reduce the index of a matrix by 1 through specific algebraic transformations, which is crucial for analyzing the anti-triangular structure.

B. Bounds on the Drazin Index

The authors derive explicit lower and upper bounds for $i(M)$ based on the indices of the diagonal block $A$ and the product $BC$ (denoted $W$):

1. Case: $ABC = BCA = 0$ (Strong Orthogonality)
   If the blocks satisfy $ABC = 0$ and $BCA = 0$, the index is tightly bounded:
   $$\max\{i(A), 2i(BC) - 1\} \leq i(M) \leq \max\{i(A), 2i(BC) - 1\} + 2$$
   In this case, a closed-form expression for $M^D$ is provided.

2. Case: $ABC = 0$ (Weaker Orthogonality)
   If only $ABC = 0$ (implying $XY=0$ in the decomposition), the bounds are slightly wider:
   $$\max\{i(A), 2i(BC) - 1\} - 1 \leq i(M) \leq i(A) + 2i(BC) + 2$$
   The paper provides a complex but explicit formula for $M^D$ involving finite sums of powers of $A^D$ and $(BC)^D$.

3. Special Cases:

   • If $A=0$ (Bipartite matrices), the bounds become $2i(BC) - 1 \leq i(M) \leq 2i(BC) + 1$.
   • If $BC$ is nonsingular, $M$ is either invertible (if $B, C$ are invertible) or has index 1.
   • If $A$ is nonsingular and $BC=0$, $i(M)$ is either 1 or 2, with a closed-form inverse provided.

C. Closed-Form Drazin Inverse

The paper provides explicit formulas for $M^D$ under various conditions. For instance, when $ABC = BCA = 0$:
$$M^D = \begin{bmatrix} A^D & (A^D)^2 B + B(CB)^D \\ C(A^D)^2 + (CB)^D C & C(A^D)^3 B \end{bmatrix}$$
(Note: The actual formula in the paper involves specific combinations of $A^D$ and $(BC)^D$ derived via the factorization method).

D. Graph-Theoretic Applications

The results are applied to directed graphs (digraphs). Specifically:

• Bipartite Digraphs: Their adjacency matrices are of the form $\begin{bmatrix} 0 & B \\ C & 0 \end{bmatrix}$. The paper refines the index bounds for these matrices.
• Linked and Double Star Digraphs: The adjacency matrices of these structures fit the anti-triangular form. The authors demonstrate that for certain weighted star graphs, the matrix is group invertible (index 1) or has a predictable Drazin index, validating the sharpness of their theoretical bounds.

4. Significance

• Theoretical Advancement: This work fills a gap in the theory of generalized inverses by moving from triangular to anti-triangular block matrices, providing the first general framework for bounding the Drazin index of such structures.
• Computational Utility: By providing explicit formulas for $M^D$ and tight bounds for $i(M)$, the paper offers practical tools for analyzing singular systems arising in control theory, Markov chains, and network analysis without needing to compute high powers of the full matrix $M$.
• Sharpness: The inclusion of numerical examples over rational numbers demonstrates that the derived bounds are sharp (i.e., the index can actually reach the lower and upper limits), confirming the optimality of the results.
• Interdisciplinary Impact: The application to graph matrices bridges linear algebra and combinatorics, offering new insights into the spectral properties and invertibility of adjacency matrices for complex network topologies.

In summary, the paper successfully generalizes the theory of Drazin indices to anti-triangular block matrices using innovative decomposition techniques, providing both theoretical bounds and constructive formulas with direct applications to graph theory.