A short note on Multilevel Toeplitz Matrices

Lei Cao; Selcuk Koyuncu

arxiv: 1906.10596 · v1 · pith:QNJ6CANZnew · submitted 2019-06-25 · 🧮 math.FA · math.OA

A short note on Multilevel Toeplitz Matrices

Lei Cao , Selcuk Koyuncu This is my paper

Pith reviewed 2026-05-25 16:00 UTC · model grok-4.3

classification 🧮 math.FA math.OA

keywords multilevel Toeplitz matricescomplex symmetric matricesunitary similaritytensor productsparity of dimensionToeplitz operators

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The pith

Any multilevel Toeplitz matrix is unitarily similar to a complex symmetric matrix via tensor products of parity-dependent unitaries.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends a one-level result to show that every multilevel Toeplitz matrix admits a unitary similarity turning it into a complex symmetric matrix. The unitary is constructed explicitly as the tensor product of the two known one-level unitaries, selected according to the parity of the dimension at each level. This construction works uniformly for any number of levels because the nested block structure of multilevel Toeplitz matrices respects the tensor product. The result also produces a new class of complex symmetric matrices that are unitarily similar to certain p-level Toeplitz matrices. A reader cares because the transformation reduces questions about these structured matrices to the symmetric case without losing the multilevel pattern.

Core claim

Chien, Liu, Nakazato, and Tam proved that every one-level Toeplitz matrix is unitarily similar to a complex symmetric matrix by one of two unitaries whose choice depends only on the parity of n. We extend this to p-level Toeplitz matrices: any such matrix is unitarily similar to a complex symmetric matrix, with the unitary obtained by taking the tensor product of the appropriate one-level unitaries for each level according to its parity. We also introduce a class of complex symmetric matrices that are unitarily similar to some p-level Toeplitz matrices.

What carries the argument

Tensor product of the two parity-dependent unitary matrices from the one-level case, chosen per level to turn the multilevel block Toeplitz structure into a complex symmetric matrix.

If this is right

Every multilevel Toeplitz matrix, regardless of the number of levels, can be transformed into complex symmetric form by a unitary that depends only on the parities of the level sizes.
The same two one-level unitaries suffice for all levels; no new unitaries need to be invented for higher levels.
A new family of complex symmetric matrices is obtained, each of which is unitarily similar to a multilevel Toeplitz matrix.
The construction is uniform: the same rule applies to any p and any choice of level dimensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The tensor-product method may extend to other nested-block structures such as multilevel Hankel or quasi-Toeplitz matrices.
Numerical linear algebra routines that exploit complex symmetry could now be applied directly to multilevel Toeplitz problems after this transformation.
The parity-based choice suggests a connection to the representation of the group (Z/2Z)^p acting on the space of multilevel matrices.

Load-bearing premise

The nested block structure of a multilevel Toeplitz matrix remains compatible with the tensor product of the one-level unitaries so that the result is complex symmetric.

What would settle it

A concrete multilevel Toeplitz matrix (for example a 2-by-2 block matrix with odd-sized blocks) for which the tensor-product unitary fails to produce a complex symmetric matrix.

read the original abstract

Chien, Liu, Nakazato, and Tam proved that all n by n classical Toeplitz matrices (one-level Toeplitz matrices) are unitarily similar to complex symmetric matrices via two types of unitary matrices and the type of the unitary matrices only depends on the parity of n. In this paper, we extend their result to multilevel Toeplitz matrices that any multilevel Toeplitz matrix is unitarily similar to a complex symmetric matrix. We provide a method to construct the unitary matrices that uniformly turn any multilevel Toeplitz matrix to a complex symmetric matrix by taking tensor products of these two types of unitary matrices for one-level Toeplitz matrices according to the parity of each level of the multilevel Toeplitz matrices. In addition, we introduce a class of complex symmetric matrices that are unitarily similar to some p-level Toeplitz matrices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper cleanly extends the one-level unitary similarity result to multilevel Toeplitz matrices via a tensor-product construction that respects the nested block structure.

read the letter

The core contribution is the uniform construction: tensor the parity-dependent unitaries from the one-level case across levels to produce a single unitary that turns any multilevel Toeplitz matrix into a complex symmetric one. The stress-test note confirms this works because symmetrizing the inner blocks first and then the block arrangement itself preserves the property under the multilevel transpose. That step is new relative to Chien et al. and rests on the block nesting rather than any extra fitting or assumption. The paper also states a class of complex symmetric matrices that arise this way, though it does little more than name them. The argument is short and direct; no circularity appears once the tensor step is granted. The only soft spot is that the new class receives almost no further properties or examples, so its utility stays implicit. This is a narrow but self-contained note for people already working with structured matrices or multilevel operators. It is not broad, but the construction is explicit enough and the logic holds up on the given reasoning. I would bring it to a reading group if the group already covers Toeplitz or block-structured operators. I would not cite it in my own work unless I needed the multilevel extension specifically. A serious editor should send it to peer review; the extension is modest but properly grounded and worth a quick check by a referee familiar with the one-level result.

Referee Report

0 major / 1 minor

Summary. The paper extends the result of Chien, Liu, Nakazato, and Tam that every one-level n×n Toeplitz matrix is unitarily similar to a complex symmetric matrix, with the unitary depending only on the parity of n. It claims that the same holds for multilevel Toeplitz matrices: any such matrix is unitarily similar to a complex symmetric matrix via a unitary constructed as the tensor product of the one-level unitaries, with the choice of each factor determined by the parity of the dimension at that level. The paper also introduces a class of complex symmetric matrices that are unitarily similar to p-level Toeplitz matrices.

Significance. If the tensor-product construction is valid, the note supplies an explicit, uniform method to produce the unitary similarity for arbitrary multilevel Toeplitz matrices. This directly leverages the one-level result and the nested block structure of multilevel Toeplitz matrices, potentially simplifying spectral or numerical analysis of such operators by reducing them to the complex-symmetric case.

minor comments (1)

The abstract and introduction refer to 'these two types of unitary matrices' without restating their explicit form or the precise parity rule from the cited one-level result; a brief reminder would improve readability for readers who have not consulted the reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and for recommending acceptance. The provided summary accurately describes the extension of the one-level result to the multilevel setting via the tensor-product construction and the additional class of complex symmetric matrices introduced in the note.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper extends a cited one-level result (Chien et al.) to multilevel Toeplitz matrices via an explicit tensor-product construction of unitary matrices chosen by parity per level. This construction is independent of the one-level proof details and does not reduce the multilevel claim to a fit, self-definition, or self-citation chain. The cited one-level result is external (different authors) and the tensor-product step is a new, verifiable algebraic argument compatible with the nested block structure. No load-bearing step collapses to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on the validity of the cited one-level result and on the algebraic compatibility of tensor products with the multilevel Toeplitz block structure.

axioms (1)

domain assumption Chien, Liu, Nakazato, and Tam result that every one-level Toeplitz matrix is unitarily similar to a complex symmetric matrix via one of two parity-dependent unitaries.
The paper explicitly extends this result; the multilevel claim inherits its truth value.

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discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

Unitary equivalence to a com- plex symmetric matrix: geometric criteria

Levon Balayan and Stephan Ramon Garcia. Unitary equivalence to a com- plex symmetric matrix: geometric criteria. Oper. Matrices, 4(1):53–76, 2010. 21

work page 2010
[2]

Toeplitz matrices are unitarily similar to symmetric matrices

Mao-Ting Chien, Jianzhen Liu, Hiroshi Nakazato, and Tin-Yau Tam . Toeplitz matrices are unitarily similar to symmetric matrices. Linear Mul- tilinear Algebra, 65(10):2131–2144, 2017

work page 2017
[3]

Poore, and Madeline K

Stephan Ramon Garcia, Daniel E. Poore, and Madeline K. Wyse. Un itary equivalence to a complex symmetric matrix: a modulus criterion. Oper. Matrices, 5(2):273–287, 2011

work page 2011
[4]

Complex symmetric oper ators and applications

Stephan Ramon Garcia and Mihai Putinar. Complex symmetric oper ators and applications. II. Trans. Amer. Math. Soc. , 359(8):3913–3931, 2007

work page 2007
[5]

Horn and Charles R

Roger A. Horn and Charles R. Johnson. Matrix analysis . Cambridge Uni- versity Press, Cambridge, second edition, 2013

work page 2013
[6]

E. E. Tyrtyshnikov and N. L. Zamarashkin. Spectra of multilevel Toeplitz matrices: advanced theory via simple matrix relationships. Linear Algebra Appl., 270:15–27, 1998. 22

work page 1998

[1] [1]

Unitary equivalence to a com- plex symmetric matrix: geometric criteria

Levon Balayan and Stephan Ramon Garcia. Unitary equivalence to a com- plex symmetric matrix: geometric criteria. Oper. Matrices, 4(1):53–76, 2010. 21

work page 2010

[2] [2]

Toeplitz matrices are unitarily similar to symmetric matrices

Mao-Ting Chien, Jianzhen Liu, Hiroshi Nakazato, and Tin-Yau Tam . Toeplitz matrices are unitarily similar to symmetric matrices. Linear Mul- tilinear Algebra, 65(10):2131–2144, 2017

work page 2017

[3] [3]

Poore, and Madeline K

Stephan Ramon Garcia, Daniel E. Poore, and Madeline K. Wyse. Un itary equivalence to a complex symmetric matrix: a modulus criterion. Oper. Matrices, 5(2):273–287, 2011

work page 2011

[4] [4]

Complex symmetric oper ators and applications

Stephan Ramon Garcia and Mihai Putinar. Complex symmetric oper ators and applications. II. Trans. Amer. Math. Soc. , 359(8):3913–3931, 2007

work page 2007

[5] [5]

Horn and Charles R

Roger A. Horn and Charles R. Johnson. Matrix analysis . Cambridge Uni- versity Press, Cambridge, second edition, 2013

work page 2013

[6] [6]

E. E. Tyrtyshnikov and N. L. Zamarashkin. Spectra of multilevel Toeplitz matrices: advanced theory via simple matrix relationships. Linear Algebra Appl., 270:15–27, 1998. 22

work page 1998