Kronecker Power Matrices

• Kronecker power matrices are defined as repeated Kronecker products that form recursively structured objects with inherent self-similarity.
• They enable exact factorization and decomposition, providing clear criteria for identifying tensor structures in both algebraic and combinatorial settings.
• Applications span coding theory, circuit complexity, and representation theory, where their scalability aids efficient performance analysis.

Kronecker power matrices are matrices of the form

$$A^{\otimes k}=\underbrace{A\otimes A\otimes\cdots\otimes A}_{k\ \text{times}},$$

obtained by repeated Kronecker product of a fixed matrix $A$. In the literature summarized here, they appear both as an explicit object of study and as a special case of broader Kronecker-product constructions. The recurring themes are recursive self-similarity, exact factorization, preservation or transformation of structural invariants under repeated tensoring, and the use of Kronecker powers as a scalable mechanism in coding theory, linear-circuit complexity, structured matrix analysis, and representation theory (Lee et al., 2011, Alman et al., 2022, Romero, 2021).

1. Definition and algebraic setting

For matrices $A_1,\dots,A_N$, associativity of the Kronecker product allows the iterated construction

$A_1\otimes A_2\otimes\cdots\otimes A_N,$

and in particular the repeated self-product

$$A^{\otimes N}=\underbrace{A\otimes A\otimes\cdots\otimes A}_{N\text{ times}}.$$

This repeated construction is explicit in the analysis of polarizing matrices, low-depth linear circuits, and exact binary factorization frameworks (Lee et al., 2011, Alman et al., 2022, Voet et al., 29 Oct 2025).

Several papers place Kronecker powers within a larger algebraic environment rather than treating them as an isolated class. One such environment is the distinction between Kronecker products and Kronecker sums. For example,

$A\oplus B=A\otimes I_2+I_2\otimes B$

and, more generally, the $k$-fold Kronecker sum

$$A^{\oplus k} = A\otimes I_n\otimes\cdots\otimes I_n +I_n\otimes A\otimes\cdots\otimes I_n +\cdots +I_n\otimes\cdots\otimes I_n\otimes A$$

appear in work on Lorentz-group representations, matrix compounds, and matrix-function decay (Larsson et al., 2021, Ofir et al., 2024, Benzi et al., 2015). This distinction is central: some papers directly analyze $A^{\otimes k}$, whereas others analyze repeated Kronecker-sum structures and note only indirect relevance to “Kronecker power matrices” (Larsson et al., 2021, Benzi et al., 2015).

The literature also emphasizes rearrangement and commutation phenomena. Shuffling matrices provide permutation operators that reorder factors in iterated Kronecker products. In the square case,

$$L(A_0\otimes A_1\otimes\cdots\otimes A_{m-1})L^{-1} = A_{\sigma^{-1}(0)}\otimes A_{\sigma^{-1}(1)}\otimes\cdots\otimes A_{\sigma^{-1}(m-1)},$$

and in the homogeneous case $A$0, the shuffling action yields an embedding

$A$1

For Kronecker powers, this means that when all factors are equal, coordinate permutations can be implemented by a structured permutation group acting on $A$2 points (D'Angeli et al., 2016).

A useful interpretive point follows from these sources. Kronecker power matrices are not merely larger copies of a base matrix; they are recursively structured tensor objects whose indexing, symmetry, and factor order can often be manipulated by exact algebraic operations. This suggests why they recur in areas where recursive scaling is essential.

2. Exact factorization, decomposition, and recognition

A major line of work studies when a matrix is exactly a Kronecker product and how repeated factorizations reveal a Kronecker power structure. For a nonzero $A$3 matrix $A$4, partitioned into $A$5 blocks of size $A$6, the criterion

$A$7

is necessary and sufficient for the existence of matrices $A$8 and $A$9 such that

$A_1,\dots,A_N$0

Here $A_1,\dots,A_N$1 is the block vec matrix, and the key identity is

$A_1,\dots,A_N$2

Thus a Kronecker product becomes an outer product after rearrangement, so the rearranged matrix has rank $A_1,\dots,A_N$3 (Ojeda, 2013).

The square-root problem specializes this factorization criterion to the second Kronecker power. If $A_1,\dots,A_N$4 is an $A_1,\dots,A_N$5 matrix, then

$A_1,\dots,A_N$6

A necessary condition is that $A_1,\dots,A_N$7 be symmetric and rank one, and this condition is sufficient over $A_1,\dots,A_N$8. Over $A_1,\dots,A_N$9, the additional condition

$A_1\otimes A_2\otimes\cdots\otimes A_N,$0

characterizes the existence of a real Kronecker square root. The square root is unique up to sign (Ojeda, 2013). A direct implication stated in the same work is that higher Kronecker powers can be tested recursively: one may first test whether a matrix is a Kronecker square, then test whether the extracted factor is itself a Kronecker square, and so on.

For binary matrices, exact factorization can be characterized directly from sparsity patterns. A binary matrix $A_1\otimes A_2\otimes\cdots\otimes A_N,$1 is decomposable if

$A_1\otimes A_2\otimes\cdots\otimes A_N,$2

with $A_1\otimes A_2\otimes\cdots\otimes A_N,$3, each factor square, and each factor size $A_1\otimes A_2\otimes\cdots\otimes A_N,$4. For fixed sizes $A_1\otimes A_2\otimes\cdots\otimes A_N,$5, the factorization of $A_1\otimes A_2\otimes\cdots\otimes A_N,$6 is unique. A length-2 factorization with dimensions $A_1\otimes A_2\otimes\cdots\otimes A_N,$7 exists if and only if the support-derived set $A_1\otimes A_2\otimes\cdots\otimes A_N,$8 satisfies

$A_1\otimes A_2\otimes\cdots\otimes A_N,$9

Equivalently, the rearranged matrix $$A^{\otimes N}=\underbrace{A\otimes A\otimes\cdots\otimes A}_{N\text{ times}}.$$0 has rank $$A^{\otimes N}=\underbrace{A\otimes A\otimes\cdots\otimes A}_{N\text{ times}}.$$1. In the specific case $$A^{\otimes N}=\underbrace{A\otimes A\otimes\cdots\otimes A}_{N\text{ times}}.$$2, the support set should recursively exhibit this Cartesian-product structure at every level (Voet et al., 29 Oct 2025).

The same framework explains how repeated self-similarity builds longer decompositions. The paper proves that compatible length-2 factorizations can be stitched into longer chains, and interprets the result through a decomposition graph whose paths encode factorization branches. In that language, a Kronecker power matrix is a particularly structured decomposable matrix whose repeated factorization manifests as a branchable path in the decomposition graph (Voet et al., 29 Oct 2025).

These results collectively establish two complementary viewpoints. One viewpoint is algebraic: rearrangement plus rank-one structure detects exact Kronecker factorization. The other is combinatorial: sparsity pattern plus recursive Cartesian-product structure detects self-similar tensor decomposition. Together they form the basic recognition theory for Kronecker power matrices.

3. Coding-theoretic invariants and polarizing matrices

In coding theory, Kronecker powers appear as the scaling mechanism for polarizing matrices. A binary $$A^{\otimes N}=\underbrace{A\otimes A\otimes\cdots\otimes A}_{N\text{ times}}.$$3 matrix

$$A^{\otimes N}=\underbrace{A\otimes A\otimes\cdots\otimes A}_{N\text{ times}}.$$4

is treated as a polarizing matrix, and its polarization behavior is quantified through the partial distances

$$A^{\otimes N}=\underbrace{A\otimes A\otimes\cdots\otimes A}_{N\text{ times}}.$$5

and

$$A^{\otimes N}=\underbrace{A\otimes A\otimes\cdots\otimes A}_{N\text{ times}}.$$6

The associated exponent is

$$A^{\otimes N}=\underbrace{A\otimes A\otimes\cdots\otimes A}_{N\text{ times}}.$$7

This exponent determines the asymptotic speed of channel polarization under successive cancellation decoding (Lee et al., 2011).

For a Kronecker product $$A^{\otimes N}=\underbrace{A\otimes A\otimes\cdots\otimes A}_{N\text{ times}}.$$8, the partial distances factor exactly: $$A^{\otimes N}=\underbrace{A\otimes A\otimes\cdots\otimes A}_{N\text{ times}}.$$9 The paper extends this recursively to arbitrary products

$A\oplus B=A\otimes I_2+I_2\otimes B$0

The exponent correspondingly becomes a weighted sum of component exponents: $A\oplus B=A\otimes I_2+I_2\otimes B$1 and more generally

$A\oplus B=A\otimes I_2+I_2\otimes B$2

These identities make large Kronecker constructions analytically transparent (Lee et al., 2011).

The repeated-power case is especially striking: $A\oplus B=A\otimes I_2+I_2\otimes B$3 Hence Kronecker powering does not improve or degrade the exponent. The large matrix $A\oplus B=A\otimes I_2+I_2\otimes B$4 retains exactly the same polarization exponent as the base matrix $A\oplus B=A\otimes I_2+I_2\otimes B$5 (Lee et al., 2011). The design principle stated in the paper is therefore to choose component matrices with high exponent, combine them via Kronecker products, and predict performance from the closed forms for partial distances and exponents.

This coding-theoretic perspective provides one of the clearest examples of why Kronecker powers matter: they scale blocklength while preserving a core asymptotic invariant. A plausible implication is that, in this setting, the value of Kronecker powering lies less in changing the qualitative polarization rate than in enabling analytically controlled length growth.

4. Circuit complexity and coverings of Kronecker powers

Another major research direction studies the complexity of computing transformations defined by Kronecker powers. For a fixed $A\oplus B=A\otimes I_2+I_2\otimes B$6 matrix $A\oplus B=A\otimes I_2+I_2\otimes B$7, the matrix $A\oplus B=A\otimes I_2+I_2\otimes B$8 has size $A\oplus B=A\otimes I_2+I_2\otimes B$9 with $k$0. A standard argument gives a depth-2 circuit of size $k$1, and depth-$k$2 size about $k$3. The paper “Smaller Low-Depth Circuits for Kronecker Powers” improves this for all fixed base matrices (Alman et al., 2022).

Its universal theorem states that for every integer $k$4, there exists

$k$5

such that for any field and any $k$6 matrix $k$7,

$k$8

The same paper gives sharper bounds in notable special cases: $k$9 for any $$A^{\oplus k} = A\otimes I_n\otimes\cdots\otimes I_n +I_n\otimes A\otimes\cdots\otimes I_n +\cdots +I_n\otimes\cdots\otimes I_n\otimes A$$0 matrix $$A^{\oplus k} = A\otimes I_n\otimes\cdots\otimes I_n +I_n\otimes A\otimes\cdots\otimes I_n +\cdots +I_n\otimes\cdots\otimes I_n\otimes A$$1,

$$A^{\oplus k} = A\otimes I_n\otimes\cdots\otimes I_n +I_n\otimes A\otimes\cdots\otimes I_n +\cdots +I_n\otimes\cdots\otimes I_n\otimes A$$2

for the Walsh-Hadamard transform $$A^{\oplus k} = A\otimes I_n\otimes\cdots\otimes I_n +I_n\otimes A\otimes\cdots\otimes I_n +\cdots +I_n\otimes\cdots\otimes I_n\otimes A$$3, and

$$A^{\oplus k} = A\otimes I_n\otimes\cdots\otimes I_n +I_n\otimes A\otimes\cdots\otimes I_n +\cdots +I_n\otimes\cdots\otimes I_n\otimes A$$4

for the disjointness matrix $$A^{\oplus k} = A\otimes I_n\otimes\cdots\otimes I_n +I_n\otimes A\otimes\cdots\otimes I_n +\cdots +I_n\otimes\cdots\otimes I_n\otimes A$$5 (Alman et al., 2022).

The technical novelty is an imbalance-aware conversion from a circuit for $$A^{\oplus k} = A\otimes I_n\otimes\cdots\otimes I_n +I_n\otimes A\otimes\cdots\otimes I_n +\cdots +I_n\otimes\cdots\otimes I_n\otimes A$$6 to a circuit for $$A^{\oplus k} = A\otimes I_n\otimes\cdots\otimes I_n +I_n\otimes A\otimes\cdots\otimes I_n +\cdots +I_n\otimes\cdots\otimes I_n\otimes A$$7. Rather than balancing a decomposition of $$A^{\oplus k} = A\otimes I_n\otimes\cdots\otimes I_n +I_n\otimes A\otimes\cdots\otimes I_n +\cdots +I_n\otimes\cdots\otimes I_n\otimes A$$8, the construction exploits imbalanced decompositions and switches between soft-balancing and hard-balancing updates while analyzing the resulting size through a random walk (Alman et al., 2022). The paper explicitly states that the new bounds provably could not be achieved using the approaches of prior work.

A related but distinct line of work studies coverings of boolean Kronecker powers. For a symmetric boolean matrix $$A^{\oplus k} = A\otimes I_n\otimes\cdots\otimes I_n +I_n\otimes A\otimes\cdots\otimes I_n +\cdots +I_n\otimes\cdots\otimes I_n\otimes A$$9, one considers coverings by rectangles and asks whether

$A^{\otimes k}$0

or analogous bounds from a chosen covering. The main synthesis theorem in the symmetric case gives, under an explicit condition involving a compact covering $A^{\otimes k}$1, a compact one-sided covering $A^{\otimes k}$2, and the parameters $A^{\otimes k}$3, $A^{\otimes k}$4, $A^{\otimes k}$5, and $A^{\otimes k}$6,

$A^{\otimes k}$7

Applied to the Kneser–Sierpinski matrices

$A^{\otimes k}$8

this yields

$A^{\otimes k}$9

improving the previously known $$L(A_0\otimes A_1\otimes\cdots\otimes A_{m-1})L^{-1} = A_{\sigma^{-1}(0)}\otimes A_{\sigma^{-1}(1)}\otimes\cdots\otimes A_{\sigma^{-1}(m-1)},$$0-type bound (Sergeev, 2022).

These complexity results show that Kronecker powers are not only algebraically recursive but also algorithmically nontrivial. Their recursive structure can be leveraged to improve circuit size and covering complexity, but only when the imbalance produced by repeated tensoring is controlled with sufficient precision.

5. Representation theory and harmonic Kronecker powers

Kronecker powers also arise in a representation-theoretic setting, where the object is no longer a matrix over a numerical field but the $$L(A_0\otimes A_1\otimes\cdots\otimes A_{m-1})L^{-1} = A_{\sigma^{-1}(0)}\otimes A_{\sigma^{-1}(1)}\otimes\cdots\otimes A_{\sigma^{-1}(m-1)},$$1-fold tensor power of a module. The paper on harmonics of $$L(A_0\otimes A_1\otimes\cdots\otimes A_{m-1})L^{-1} = A_{\sigma^{-1}(0)}\otimes A_{\sigma^{-1}(1)}\otimes\cdots\otimes A_{\sigma^{-1}(m-1)},$$2 studies the decomposition of

$$L(A_0\otimes A_1\otimes\cdots\otimes A_{m-1})L^{-1} = A_{\sigma^{-1}(0)}\otimes A_{\sigma^{-1}(1)}\otimes\cdots\otimes A_{\sigma^{-1}(m-1)},$$3

the $$L(A_0\otimes A_1\otimes\cdots\otimes A_{m-1})L^{-1} = A_{\sigma^{-1}(0)}\otimes A_{\sigma^{-1}(1)}\otimes\cdots\otimes A_{\sigma^{-1}(m-1)},$$4-fold tensor power of the harmonic or coinvariant module of the symmetric group (Romero, 2021).

The starting point is the harmonic module

$$L(A_0\otimes A_1\otimes\cdots\otimes A_{m-1})L^{-1} = A_{\sigma^{-1}(0)}\otimes A_{\sigma^{-1}(1)}\otimes\cdots\otimes A_{\sigma^{-1}(m-1)},$$5

with $$L(A_0\otimes A_1\otimes\cdots\otimes A_{m-1})L^{-1} = A_{\sigma^{-1}(0)}\otimes A_{\sigma^{-1}(1)}\otimes\cdots\otimes A_{\sigma^{-1}(m-1)},$$6. Using Chevalley’s theorem,

$$L(A_0\otimes A_1\otimes\cdots\otimes A_{m-1})L^{-1} = A_{\sigma^{-1}(0)}\otimes A_{\sigma^{-1}(1)}\otimes\cdots\otimes A_{\sigma^{-1}(m-1)},$$7

and therefore

$$L(A_0\otimes A_1\otimes\cdots\otimes A_{m-1})L^{-1} = A_{\sigma^{-1}(0)}\otimes A_{\sigma^{-1}(1)}\otimes\cdots\otimes A_{\sigma^{-1}(m-1)},$$8

This reduces the decomposition problem for $$L(A_0\otimes A_1\otimes\cdots\otimes A_{m-1})L^{-1} = A_{\sigma^{-1}(0)}\otimes A_{\sigma^{-1}(1)}\otimes\cdots\otimes A_{\sigma^{-1}(m-1)},$$9 to the $A$00-character of a polynomial ring with $A$01 sets of variables (Romero, 2021).

The Frobenius characteristic of $A$02 is expressed as

$A$03

After specializing $A$04, the multiplicity formula becomes

$A$05

The paper then derives explicit combinatorial formulas using a generalized $A$06 statistic on permutations and tableaux. Its main theorem states that for every partition $A$07,

$A$08

This gives an explicit decomposition of arbitrary Kronecker powers of harmonics (Romero, 2021).

This body of work broadens the meaning of “Kronecker power matrices.” In linear algebra and coding theory, the object is a repeated tensor power of a fixed matrix. In this representation-theoretic context, the same repeated-tensor idea governs the decomposition of modules and characters. The common structural feature is the same: repeated self-tensoring creates a recursively scalable object whose decomposition can still be described explicitly.

6. Related constructions, extensions, and common distinctions

Several papers are relevant to Kronecker power matrices precisely because they clarify what is adjacent to, but not identical with, repeated self-Kronecker products.

One example is the Lorentz-group spinor map. The paper on $A$09 and the restricted Lorentz group derives the formula

$A$10

for the induced $A$11 Lorentz transformation. The derivation uses vectorization, the identity

$A$12

and the Kronecker sum identity

$A$13

However, the paper explicitly does not discuss repeated Kronecker products or tensor powers such as

$A$14

Its Kronecker usage is always of the form $A$15, $A$16, or $A$17 (Larsson et al., 2021). This distinction is important because it separates single-product representation-theoretic constructions from true Kronecker powers.

A second adjacent line is the theory of matrix compounds. The paper on multiplicative and additive compounds shows that compounds can be written as projections of Kronecker powers and Kronecker sums: $A$18 The identities

$A$19

show that repeated Kronecker constructions act as an ambient space from which compound matrices are extracted (Ofir et al., 2024). In this sense, Kronecker powers are not always the final object of interest; they may instead be a computational or conceptual lift.

A third adjacent line concerns structured covariance estimation. In the matrix normal model

$A$20

the covariance of $A$21 has Kronecker form $A$22, and the parameter dimension is reduced because

$A$23

The paper identifies a special regime

$A$24

in which the Kronecker covariance MLE has algebraic degree one and admits the rational closed form

$A$25

This is not a Kronecker power result, but it illustrates the broader methodological role of Kronecker structure in algebraic statistics (Drton et al., 2024).

Finally, a substantial body of work concerns Kronecker sums rather than Kronecker powers. For

$A$26

the inverse $A$27 exhibits non-monotone decay governed by grid geometry rather than one-dimensional index distance. The same perspective extends to matrix functions of

$A$28

for which factorized exponential identities such as

$A$29

yield product decay structure in coordinate directions (Canuto et al., 2013, Benzi et al., 2015). These papers are often associated with tensor-structured operators, but they explicitly emphasize that the exact factorization results apply to sums, not products.

The cumulative lesson is that the phrase “Kronecker power matrices” sits inside a wider ecosystem of Kronecker-based constructions. Some works study $A$30 directly; some use it as an intermediate representation; others clarify by contrast that the relevant structure is instead a single Kronecker product or a Kronecker sum. For arXiv-level literature, this distinction is not terminological decoration but a genuine mathematical boundary.