Quaternion Toeplitz Matrices

• Quaternion Toeplitz matrices are matrices over the quaternion field with constant-diagonal entries and require distinct left/right multiplication due to noncommutativity.
• They are analyzed using displacement operators and quaternion-valued generating functions which lead to block diagonalization and efficient FFT-based inversion methods.
• Recent studies extend the theory with Hermitian symbols, multilevel block structures, and preconditioned solvers for applications in quaternion signal processing.

Quaternion Toeplitz matrices are matrices over the quaternion skew field $\mathbb H$ whose entries are constant along diagonals, typically written $T_n=[t_{j-i}]_{i,j=0}^{n-1}$ or, with an equivalent indexing convention, $T_{p,q}=a_{p-q}$. Their theory differs sharply from the real and complex settings because quaternion multiplication is noncommutative, left and right actions must be distinguished, and Fourier methods produce block structures rather than scalar diagonalizations in the generic case. Current research treats the subject from several complementary viewpoints: algebraic closure and maximal left algebras (Yagoub, 2024), displacement-based structure and normality (Khan et al., 5 Oct 2025), Hermitian generating functions and Szegő-type asymptotics (Lin et al., 21 Apr 2025), block diagonalization of associated quaternion circulants (Pan et al., 2023), and multilevel block Toeplitz distribution theory under a single-axis assumption (Lailoune et al., 27 Nov 2025).

1. Algebraic setting and basic definitions

A quaternion $q\in\mathbb H$ has the form

$q=q_0+i q_1+j q_2+k q_3,$

with $i^2=j^2=k^2=ijk=-1$, and multiplication rules such as $ij=k=-ji$. This noncommutativity is not a peripheral complication: it determines how Toeplitz systems are posed, how Fourier coefficients are defined, and which algebraic identities survive.

A quaternion Toeplitz matrix is determined by its first row and first column. In one convention,

$T_{p,q}=a_{p-q},\qquad 0\le p,q\le n-1,$

while another writes

$T_n=[t_{j-i}]_{i,j=0}^{n-1}.$

Both encode the same constant-diagonal structure. For linear systems, the left action convention is essential: $(Tx)_p=\sum_q T_{p,q}x_q.$ Because $T_n=[t_{j-i}]_{i,j=0}^{n-1}$0 is noncommutative, one cannot interchange the order of matrix entries and vector components without changing the problem (Pan et al., 2023).

Hermitian quaternion Toeplitz matrices satisfy

$T_n=[t_{j-i}]_{i,j=0}^{n-1}$1

For Hermitian $T_n=[t_{j-i}]_{i,j=0}^{n-1}$2, positive definiteness is expressed through the real part,

$T_n=[t_{j-i}]_{i,j=0}^{n-1}$3

which is the notion used for autocovariance Toeplitz matrices arising in quaternion signal processing (Pan et al., 2023).

A structural characterization uses displacement operators. If $T_n=[t_{j-i}]_{i,j=0}^{n-1}$4 denotes the unilateral shift, then a matrix $T_n=[t_{j-i}]_{i,j=0}^{n-1}$5 is Toeplitz if and only if

$T_n=[t_{j-i}]_{i,j=0}^{n-1}$6

so quaternion Toeplitz matrices have displacement rank at most $T_n=[t_{j-i}]_{i,j=0}^{n-1}$7 with respect to $T_n=[t_{j-i}]_{i,j=0}^{n-1}$8. This formulation underlies product identities, commutator formulas, and normality criteria. The same work shows that the standard complex block representation

$T_n=[t_{j-i}]_{i,j=0}^{n-1}$9

is an injective ring homomorphism preserving sums, products, adjoints, and Toeplitz structure (Khan et al., 5 Oct 2025).

2. Hermitian symbols and quaternion-valued generating functions

For Hermitian quaternion Toeplitz matrices, a symbol calculus can be built from quaternion-valued generating functions. Fix a pure unit quaternion $T_{p,q}=a_{p-q}$0 with $T_{p,q}=a_{p-q}$1, and define Fourier coefficients by right multiplication of the exponential: $T_{p,q}=a_{p-q}$2 The multiplication order is part of the definition, since $T_{p,q}=a_{p-q}$3 is noncommutative. The resulting Toeplitz matrix is $T_{p,q}=a_{p-q}$4 (Lin et al., 21 Apr 2025).

A central structural result is that a Hermitian quaternion Toeplitz matrix is not generated by an arbitrary quaternion-valued symbol. If $T_{p,q}=a_{p-q}$5 is Hermitian, then the symbol decomposes as

$T_{p,q}=a_{p-q}$6

where $T_{p,q}=a_{p-q}$7 is real-valued almost everywhere and $T_{p,q}=a_{p-q}$8 is odd. Equivalently, the symbol is the sum of a real-valued function and an odd pure-imaginary quaternion-valued function orthogonal to the chosen axis $T_{p,q}=a_{p-q}$9. This is a genuine quaternionic phenomenon: the complex Hermitian Toeplitz case is generated by real-valued symbols only (Lin et al., 21 Apr 2025).

The same theory passes through a $q\in\mathbb H$0 complex block symbol

$q\in\mathbb H$1

which is Hermitian almost everywhere. Its two eigenvalue functions, denoted $q\in\mathbb H$2 and $q\in\mathbb H$3, control the extremal behavior and asymptotic distribution of the eigenvalues of $q\in\mathbb H$4. In particular,

$q\in\mathbb H$5

and a quaternion version of the Grenander–Szegő theorem states that the limiting eigenvalue distribution is the average of the two eigenvalue branches of $q\in\mathbb H$6, rather than a single scalar symbol as in the classical complex case (Lin et al., 21 Apr 2025).

This framework also yields sufficient positivity conditions. If almost everywhere

$q\in\mathbb H$7

and $q\in\mathbb H$8, with strict inequality on a set of positive measure, then $q\in\mathbb H$9 is Hermitian positive definite for all $q=q_0+i q_1+j q_2+k q_3,$0. In quaternion covariance problems, this links spectral assumptions on the symbol directly to solvability and conditioning of Toeplitz systems (Lin et al., 21 Apr 2025).

3. Circulant companions, DQFT, and block diagonalization

The most important computational companion of a quaternion Toeplitz matrix is a quaternion circulant matrix. For

$q=q_0+i q_1+j q_2+k q_3,$1

the classical complex expectation would be diagonalization by a Fourier matrix. That expectation fails in general over quaternions. A central impossibility result states that a general quaternion circulant matrix cannot be diagonalized by any discrete quaternion Fourier transform $q=q_0+i q_1+j q_2+k q_3,$2 associated with a unit pure quaternion $q=q_0+i q_1+j q_2+k q_3,$3. Instead, one obtains

$q=q_0+i q_1+j q_2+k q_3,$4

where $q=q_0+i q_1+j q_2+k q_3,$5 is the permutation matrix that flips frequency pairs $q=q_0+i q_1+j q_2+k q_3,$6. The terms involving $q=q_0+i q_1+j q_2+k q_3,$7 and $q=q_0+i q_1+j q_2+k q_3,$8 obstruct scalar diagonalization and force coupling of the frequencies $q=q_0+i q_1+j q_2+k q_3,$9 and $i^2=j^2=k^2=ijk=-1$0 (Pan et al., 2023).

After a suitable permutation $i^2=j^2=k^2=ijk=-1$1, this transformed matrix becomes block diagonal with one-by-one blocks at the special frequencies and two-by-two blocks for paired frequencies. For odd $i^2=j^2=k^2=ijk=-1$2, there is one $i^2=j^2=k^2=ijk=-1$3 block and $i^2=j^2=k^2=ijk=-1$4 blocks of size $i^2=j^2=k^2=ijk=-1$5; for even $i^2=j^2=k^2=ijk=-1$6, there are two $i^2=j^2=k^2=ijk=-1$7 blocks and $i^2=j^2=k^2=ijk=-1$8 blocks of size $i^2=j^2=k^2=ijk=-1$9. This is the basic frequency-domain structure behind inversion and linear solves. The inverse of an invertible quaternion circulant is assembled by inverting the scalar and $ij=k=-ji$0 blocks and transforming back, with overall complexity dominated by quaternion FFT operations and therefore $ij=k=-ji$1 (Pan et al., 2023).

This block viewpoint clarifies a common misconception. The discrete quaternion Fourier transform behaves like the complex FFT only in special cases. If the entries of the circulant lie entirely in one imaginary plane, such as the $ij=k=-ji$2-plane, the obstruction terms vanish and exact diagonalization by $ij=k=-ji$3 is recovered. In the generic Hamiltonian case, however, noncommutativity prevents full diagonalization (Pan et al., 2023).

For Hermitian quaternion Toeplitz systems, this obstruction is partly circumvented by structure-preserving circulant approximants. Strang’s circulant preconditioners, constructed from the central diagonals of a Hermitian Toeplitz matrix, can be diagonalized by discrete quaternion Fourier transform matrices in block $ij=k=-ji$4 form, whereas general quaternion circulants cannot. This distinction is one of the main reasons such preconditioners are effective in Hermitian Toeplitz solvers (Lin et al., 21 Apr 2025).

4. Spectral distribution in the multilevel and block setting

The one-level theory extends to quaternion block multilevel Toeplitz sequences under a single-axis assumption. Fix an imaginary unit $ij=k=-ji$5 and write quaternion matrix-valued symbols as

$ij=k=-ji$6

with $ij=k=-ji$7 taking values in the complex slice $ij=k=-ji$8. The single-axis hypothesis ensures that Fourier monomials $ij=k=-ji$9 commute with the slice-valued coefficients, which makes the Fourier calculus coherent and allows transfer of complex Toeplitz results through symplectic embedding (Lailoune et al., 27 Nov 2025).

In this setting, one distinguishes left, right, and sandwich Fourier coefficients. For a partition $T_{p,q}=a_{p-q},\qquad 0\le p,q\le n-1,$0,

$T_{p,q}=a_{p-q},\qquad 0\le p,q\le n-1,$1

and the resulting multilevel Toeplitz matrix is formed from these coefficients. A right-kernel reduction shows that left and sandwich formulations can be transported to a right-kernel problem through a reflection of the $T_{p,q}=a_{p-q},\qquad 0\le p,q\le n-1,$2-component. This identifies the kernel choice as a structural, rather than purely notational, issue in quaternion Toeplitz analysis (Lailoune et al., 27 Nov 2025).

The corresponding embedded complex symbol is a $T_{p,q}=a_{p-q},\qquad 0\le p,q\le n-1,$3 block matrix. For example,

$T_{p,q}=a_{p-q},\qquad 0\le p,q\le n-1,$4

with analogous formulas for $T_{p,q}=a_{p-q},\qquad 0\le p,q\le n-1,$5 and $T_{p,q}=a_{p-q},\qquad 0\le p,q\le n-1,$6. The singular-value distribution of the quaternion Toeplitz sequence is then governed by the singular values of the embedded symbol. If $T_{p,q}=a_{p-q},\qquad 0\le p,q\le n-1,$7, the sequence is distributed in the sense of Weyl with symbol $T_{p,q}=a_{p-q},\qquad 0\le p,q\le n-1,$8; if the square case is Hermitian, the eigenvalue distribution is likewise determined by the eigenvalues of $T_{p,q}=a_{p-q},\qquad 0\le p,q\le n-1,$9. For bounded non-Hermitian symbols, the same conclusion holds under the Tilli separation conditions $T_n=[t_{j-i}]_{i,j=0}^{n-1}.$0 and $T_n=[t_{j-i}]_{i,j=0}^{n-1}.$1 connected (Lailoune et al., 27 Nov 2025).

Hermitian character also ունի a symbol criterion in the multilevel framework: $T_n=[t_{j-i}]_{i,j=0}^{n-1}.$2 is Hermitian for every $T_n=[t_{j-i}]_{i,j=0}^{n-1}.$3 if and only if $T_n=[t_{j-i}]_{i,j=0}^{n-1}.$4 is essentially Hermitian and

$T_n=[t_{j-i}]_{i,j=0}^{n-1}.$5

almost everywhere. Under this condition, all eigenvalues lie in the essential range of the embedded symbol, and if the lower spectral bound $T_n=[t_{j-i}]_{i,j=0}^{n-1}.$6 is positive, the whole sequence is uniformly Hermitian positive definite (Lailoune et al., 27 Nov 2025).

5. Left algebras, boundary relations, and normality

The full set $T_n=[t_{j-i}]_{i,j=0}^{n-1}.$7 of quaternion Toeplitz matrices is a left subspace of $T_n=[t_{j-i}]_{i,j=0}^{n-1}.$8, but it is not closed under multiplication in general. This failure is one of the basic differences from many familiar complex Toeplitz subclasses. A product criterion identifies exactly when two quaternion Toeplitz matrices multiply to another Toeplitz matrix, and if the entries of two Toeplitz matrices commute pairwise, then the matrices themselves commute (Yagoub, 2024).

To recover algebraic closure, one imposes boundary relations. For a quaternion subalgebra $T_n=[t_{j-i}]_{i,j=0}^{n-1}.$9 and parameters $(Tx)_p=\sum_q T_{p,q}x_q.$0, the family

$(Tx)_p=\sum_q T_{p,q}x_q.$1

forms a left algebra in $(Tx)_p=\sum_q T_{p,q}x_q.$2. This construction subsumes several classical-looking subclasses. If $(Tx)_p=\sum_q T_{p,q}x_q.$3, one obtains the left algebra of quaternion circulant matrices. If $(Tx)_p=\sum_q T_{p,q}x_q.$4, one gets upper triangular quaternion Toeplitz matrices; if $(Tx)_p=\sum_q T_{p,q}x_q.$5, lower triangular quaternion Toeplitz matrices. Diagonal Toeplitz matrices are contained in every $(Tx)_p=\sum_q T_{p,q}x_q.$6 (Yagoub, 2024).

These algebras admit sharp classification criteria. Two parameter pairs produce the same algebra if and only if

$(Tx)_p=\sum_q T_{p,q}x_q.$7

Moreover,

$(Tx)_p=\sum_q T_{p,q}x_q.$8

Thus maximality is controlled by a commutant condition rather than by a symbol or spectral argument (Yagoub, 2024).

A different line of work addresses normality. Over Hamilton’s quaternions, the paper on fundamental properties develops displacement formulas and examples of normal and non-normal Toeplitz matrices, but the complete classification is obtained only for Toeplitz matrices with entries in Segre’s commutative quaternion algebra, not in $(Tx)_p=\sum_q T_{p,q}x_q.$9. In that commutative algebra, normality is defined with respect to one of three principal adjoints $T_n=[t_{j-i}]_{i,j=0}^{n-1}$00, and the main theorem states that

$T_n=[t_{j-i}]_{i,j=0}^{n-1}$01

if and only if, for all $T_n=[t_{j-i}]_{i,j=0}^{n-1}$02,

$T_n=[t_{j-i}]_{i,j=0}^{n-1}$03

This yields a complete characterization of all normal Toeplitz matrices having entries commutative quaternions (Khan et al., 5 Oct 2025).

6. Solvers, applications, and present limitations

Quaternion Toeplitz systems arise naturally in quaternion signal processing, especially from covariance sequences of stationary quaternion signals and from linear prediction. In that context, the symbol has the form

$T_n=[t_{j-i}]_{i,j=0}^{n-1}$04

which automatically fits the Hermitian-symbol structure described above. This connects abstract symbol theory directly with signal-processing Toeplitz systems (Lin et al., 21 Apr 2025).

The dominant computational strategy is to replace or approximate Toeplitz matrices by structured circulants. For general Toeplitz systems $T_n=[t_{j-i}]_{i,j=0}^{n-1}$05, one embeds $T_n=[t_{j-i}]_{i,j=0}^{n-1}$06 into a larger quaternion circulant $T_n=[t_{j-i}]_{i,j=0}^{n-1}$07, solves in the transform domain block by block, and extracts the leading components. For Hermitian positive definite Toeplitz matrices, preconditioned conjugate gradient with quaternion circulant preconditioners is the preferred route. The block diagonalization of circulants supplies $T_n=[t_{j-i}]_{i,j=0}^{n-1}$08 preconditioner applications because the cost is dominated by quaternion FFTs and constant-size block inversions (Pan et al., 2023).

For Hermitian systems generated by quaternion-valued functions, Strang’s circulant preconditioners enjoy provable spectral clustering. If $T_n=[t_{j-i}]_{i,j=0}^{n-1}$09 continuously on $T_n=[t_{j-i}]_{i,j=0}^{n-1}$10, then for sufficiently large $T_n=[t_{j-i}]_{i,j=0}^{n-1}$11, the spectrum of $T_n=[t_{j-i}]_{i,j=0}^{n-1}$12 is bounded away from zero and all but finitely many eigenvalues lie in $T_n=[t_{j-i}]_{i,j=0}^{n-1}$13. The corresponding PCG convergence is superlinear after the finite outlier phase, with per-iteration cost $T_n=[t_{j-i}]_{i,j=0}^{n-1}$14 (Lin et al., 21 Apr 2025).

Reported numerical behavior is consistent with this theory. For exact covariance matrices, PCG with Strang’s circulant preconditioner typically converges in $T_n=[t_{j-i}]_{i,j=0}^{n-1}$15–$T_n=[t_{j-i}]_{i,j=0}^{n-1}$16 iterations, whereas the unpreconditioned method needs tens to over a hundred iterations. For sampled covariance matrices, the reported iteration ranges are $T_n=[t_{j-i}]_{i,j=0}^{n-1}$17 to $T_n=[t_{j-i}]_{i,j=0}^{n-1}$18 for the preconditioned method and $T_n=[t_{j-i}]_{i,j=0}^{n-1}$19 to $T_n=[t_{j-i}]_{i,j=0}^{n-1}$20 without preconditioning, with similar residuals and faster wall-clock time for the circulant approach (Lin et al., 21 Apr 2025).

The multilevel theory extends these ideas to block Toeplitz sequences with continuous and $T_n=[t_{j-i}]_{i,j=0}^{n-1}$21 symbols. Quaternion block multilevel circulants form an approximating class of sequences for quaternion block multilevel Toeplitz matrices, and the resulting FFT-based preconditioners support fast Krylov methods in large systems. Numerical experiments on two-level block Toeplitz matrices with Hermitian and non-Hermitian symbols show that empirical eigenvalue and singular-value quantiles increasingly match the symbol quantiles as $T_n=[t_{j-i}]_{i,j=0}^{n-1}$22 grows, with agreement already good at moderate sizes (Lailoune et al., 27 Nov 2025).

Several limitations remain explicit in the current literature. Ill-conditioned $T_n=[t_{j-i}]_{i,j=0}^{n-1}$23 or $T_n=[t_{j-i}]_{i,j=0}^{n-1}$24 frequency blocks can destabilize inversion; Toeplitz-to-circulant embedding introduces wrap-around error; and careless mixing of left and right multiplication leads to incorrect algorithms. In the multilevel setting, the single-axis assumption is critical for commutation with Fourier kernels, and extending the theory beyond that regime is identified as a major open direction. A further open divide separates the Hamiltonian case, where noncommutativity obstructs a full normality classification, from the commutative quaternion setting, where such a classification is available (Pan et al., 2023, Lailoune et al., 27 Nov 2025, Khan et al., 5 Oct 2025).