Kronecker differences

Over the real numbers, the Kronecker sum is the unique operation on matrices which exponentiates to the Kronecker product. Kronecker quotients provide an algebraic view of decompositions of matrices in terms of Kronecker products. This article explores families of operations, Kronecker differences, which are a kind of “inverse” for Kronecker sums. The correspondence between Kronecker differences and Kronecker quotients is explored. Furthermore, we show that a certain class of Kronecker differences may be characterized by families of matrices with these families again being expressed as Kronecker products. This approach provides a different “nonlinear” view towards tensor decomposition.

💡 Research Summary

The paper introduces “Kronecker differences,” an operation that serves as a kind of inverse to the Kronecker sum, and systematically relates it to the already‑studied Kronecker quotient. After recalling the Kronecker product ⊗ and the right Kronecker quotient ⊘ (defined by (A⊗B)⊘B = A), the authors define a Kronecker difference ⊖ by the identity (A⊕B)⊖B = A, where the Kronecker sum is A⊕B = A⊗I_n + I_m⊗B. Crucially, they show that any Kronecker difference can be generated from a Kronecker quotient via the simple formula

 M⊖B = (M – I_m⊗B) ⊘ I_n.

Thus the two families of operations are in one‑to‑one correspondence.

The notion of uniformity is then introduced. A uniform Kronecker quotient satisfies (A⊗C)⊘B = A⊗(C⊘B) together with linearity in the first argument; analogously a uniform Kronecker difference satisfies (A⊕C)⊖B = A⊕(C⊖B) and is linear in both arguments. Proposition 1 proves that a uniform quotient always induces a uniform difference, and a series of propositions (2, 3) translate basic algebraic properties of the sum (transpose, trace, scalar multiplication, associativity, commutator, exponential) into corresponding statements for the difference. For example, (A⊖B)ᵀ = Aᵀ⊖Bᵀ and tr(A⊖B) = (1/n)(tr A – m tr B).

The heart of the paper concerns linear Kronecker differences, i.e., those satisfying the two linearity conditions (D3) and (D4). In this case the map δ(A,B) := A⊖B can be regarded as a bilinear map from (F^{m×n} × F^{n}) to F^{m}. The authors prove (Proposition 3, Theorem 1) that every such linear δ admits a canonical tensor representation

 δ(A,B) = tr_{12}!\bigl(αᵀ (A⊗I_m – I_m⊗B⊗I_m)\bigr),

where tr_{12} denotes the partial trace over the first two tensor factors and α ∈ F^{m×n×m} is a third‑order tensor. Moreover, α can be decomposed as

 α = Σ_{i,j} E_{ij} ⊗ E_{11} ⊗ E_{ji} + γ,

with γ ∈ F^{m×n×m} satisfying tr₂(γ) = 0. The first term is a fixed “canonical” part, while γ encodes the remaining degrees of freedom of the difference operation. When γ = 0 the difference reduces to a simple partial‑trace of the matrix A – I_m⊗B, i.e. the operation is essentially “take the block‑wise trace after subtracting the Kronecker‑product of the identity with B.” Theorem 2 extends this representation to fields whose characteristic does not divide n, inserting a factor 1/n in the canonical part.

Lemmas 2 and 3 show that many of the algebraic identities for ⊖ can be reduced to statements involving the zero matrix, confirming that the canonical form captures the essential behavior. The paper also discusses how the difference interacts with the vectorized Sylvester equation (B X + X Aᵀ = Y), since vec(Y) = (A⊕B) vec(X). Using the difference, one can rewrite the solution process as a “difference” operation applied to the right‑hand side, offering a new perspective on solving such matrix equations.

In the concluding remarks the authors point out that the canonical representation reveals a non‑linear aspect of Kronecker differences: the free tensor γ provides a non‑linear correction to the otherwise linear trace‑based operation. This suggests that Kronecker differences could serve as building blocks for novel tensor‑decomposition schemes that go beyond traditional linear models (e.g., CP or Tucker). Potential applications include signal processing, quantum information, and any domain where Kronecker‑structured linear systems appear.

Overall, the paper establishes a rigorous algebraic framework for Kronecker differences, connects them tightly to Kronecker quotients, characterizes the uniform and linear subclasses, and provides a tensor‑based canonical form that both clarifies the structure of these operations and opens avenues for further research in nonlinear tensor analysis.