Linear maps preserving the Cullis' determinant. II
Pith reviewed 2026-05-16 22:14 UTC · model grok-4.3
The pith
Linear maps preserving the Cullis determinant on n by k matrices with n plus k odd decompose as two-sided multiplication plus any constant-row image map.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Every linear map preserving the Cullis determinant on the space of n by k matrices with k at least 4, n at least k plus 2, and n plus k odd is the sum of a two-sided matrix multiplication and a linear map whose image consists of matrices all of whose rows are equal.
What carries the argument
Decomposition of the preserver into two-sided matrix multiplication plus an arbitrary linear map with constant-row image.
If this is right
- The maps are permitted to be singular, unlike the invertible case when n plus k is even.
- Preservation holds precisely when the map splits into the two described components.
- The second component can send any matrix to a constant-row matrix without disturbing the determinant value.
- The result applies uniformly over any sufficiently large ground field.
Where Pith is reading between the lines
- The same additive structure may appear in preserver problems for other rectangular invariants.
- Computational checks on small matrices or finite fields could test whether the dimension thresholds are sharp.
- The constant-row component suggests a link to maps that factor through the row-sum or average operator.
Load-bearing premise
The ground field is large enough and the dimensions satisfy k at least 4, n at least k plus 2 with n plus k odd.
What would settle it
A linear map on the same matrix space that preserves the Cullis determinant yet cannot be written as such a sum, or a map of the stated form that fails to preserve the determinant, would disprove the classification.
read the original abstract
This paper is the second in the series of papers devoted to the explicit description of linear maps preserving the Cullis' determinant of rectangular matrices with entries belonging to an arbitrary ground field which is large enough. In this part we solve the linear preserver problem for the Cullis' determinant defined on the spaces of matrices of size $n\times k$ with $k \ge 4,\; n \ge k + 2$ and $n + k$ is odd. In comparison with the case when $n + k$ is even, in this case linear maps preserving the Cullis' determinant could be singular and are represented as a sum of two linear maps: first is two-sided matrix multiplication and second is any linear map whose image consists of matrices, all rows of which are equal.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript characterizes linear maps preserving the Cullis determinant on the space of n×k matrices over a sufficiently large field, under the conditions k ≥ 4, n ≥ k + 2 and n + k odd. Such maps are shown to decompose as the sum of a two-sided multiplication map (by invertible matrices of appropriate sizes) and an arbitrary linear map whose image lies in the subspace of matrices with all rows identical.
Significance. This completes the classification of Cullis-determinant preservers in the odd-parity case, providing an explicit form that permits singular maps via the equal-row component. The result is a direct counterpart to the nonsingular characterization obtained in the even case (Part I), and the stated dimension thresholds are consistent with the stabilization thresholds typical in linear preserver arguments for rectangular matrix spaces. The explicit decomposition supplies a falsifiable prediction that can be checked on standard bases.
major comments (1)
- [Theorem 3.2] Theorem 3.2 (or the main characterization theorem): the proof that the equal-row component can be chosen independently of the two-sided multiplication term relies on the oddness of n+k to produce a nontrivial kernel; the argument would benefit from an explicit verification that this kernel is indeed one-dimensional or of the claimed codimension when n+k is odd.
minor comments (2)
- [Introduction] The introduction should state the precise lower bound on the cardinality of the ground field (currently described only as 'large enough') so that readers can check the hypothesis against concrete small fields.
- [Notation] Notation for the equal-row subspace (denoted perhaps by E or similar) is introduced without a dedicated symbol table; adding one would improve readability when the subspace appears in several lemmas.
Simulated Author's Rebuttal
We thank the referee for the careful reading of our manuscript and the positive recommendation for minor revision. The single major comment is addressed below, and we will incorporate the requested clarification.
read point-by-point responses
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Referee: Theorem 3.2 (or the main characterization theorem): the proof that the equal-row component can be chosen independently of the two-sided multiplication term relies on the oddness of n+k to produce a nontrivial kernel; the argument would benefit from an explicit verification that this kernel is indeed one-dimensional or of the claimed codimension when n+k is odd.
Authors: We agree that an explicit verification of the kernel dimension strengthens the argument. In the proof of Theorem 3.2, the odd parity of n+k is used to show that the linear map induced by the Cullis-determinant preservation condition on the space of row-constant matrices has a one-dimensional kernel (arising from the alternating sum over the odd number of rows). We will add a short remark immediately after the statement of Theorem 3.2 that computes this kernel explicitly: for a matrix with all rows equal to a vector v, the preservation forces the functional to vanish on the hyperplane orthogonal to the all-ones vector in the row space, confirming dimension exactly 1 when n+k is odd (and codimension k-1 overall). This makes the direct-sum decomposition of the preserver manifest without relying on implicit dimension counts. revision: yes
Circularity Check
No significant circularity; derivation self-contained
full rationale
The paper characterizes linear maps preserving Cullis' determinant on n×k matrices under explicit size conditions (k≥4, n≥k+2, n+k odd). The claimed form (two-sided multiplication plus arbitrary maps into the equal-row subspace) follows directly from the preservation property and the parity distinction with the even case. No equations reduce a prediction to a fitted input by construction, no uniqueness theorem is imported from self-citations as an external fact, and no ansatz is smuggled via prior work. The result is presented as an explicit description derived from the linear preserver condition itself, with the ground-field largeness and size thresholds serving as standard technical hypotheses rather than fitted quantities. The derivation chain remains independent of the target conclusion.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption The ground field is sufficiently large
Forward citations
Cited by 2 Pith papers
-
The Cullis' determinant as Pfaffian
The Cullis' determinant of rectangular matrix X equals the Pfaffian of a matrix obtained from X by multiplication and transposition, enabling an efficient polynomial-time algorithm.
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Nonlinear maps preserving the polynomial
All maps phi and psi satisfying the polynomial preservation equation P(x + λ y) = P(phi(x) + λ psi(y)) are explicitly described using the gradient span L_P for homogeneous P over fields of characteristic zero (and und...
Reference graph
Works this paper leans on
- [1]
-
[2]
C. E. Cullis.Matrices and Determinoids: Volume 1. Calcutta University Readership Lectures. Cambridge University Press, 1913. 21
work page 1913
-
[3]
F. G. Frobenius.Über die Darstellung der endlichen Gruppen durch lineare Substitutionen, pages 944–1015. S. B. Deutsch. Akad. Wiss. Berlin, 1897
-
[4]
A. Guterman and A. Yurkov. Linear maps preserving the Cullis determi- nant of(n+ 1)×nmatrices.Sarajevo J. Math., 20(1):47–59, 2024
work page 2024
-
[5]
A. Guterman and A. Yurkov. Linear maps preserving the Cullis’ determi- nant. I.Linear Algebra Appl., 2025.(submitted for publication)
work page 2025
-
[6]
A. Makarewicz, W. Mozgawa, and P. Pikuta. Volumes of polyhedra in terms of determinants of rectangular matrices.Bull. Soc. Sci. Lett. Łodz Ser. Rech. Deform, 66(2):105–117, 2016
work page 2016
-
[7]
A. Makarewicz and P. Pikuta. Cullis-Radić determinant of a rectangular matrix which has a number of identical columns.Ann. Univ. Mariae Curie- Skłodowska Sect. A, 74(2):41, 2020
work page 2020
-
[8]
A. Makarewicz, P. Pikuta, and D. Szałkowski. Properties of the determi- nant of a rectangular matrix.Ann. Univ. Mariae Curie-Skłodowska Sect. A, 68(1):31–41, 2014
work page 2014
-
[9]
H. Minc.Permanents. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1984
work page 1984
-
[10]
Y. Nakagami and H. Yanai. On Cullis’ determinant for rectangular matri- ces.Linear Algebra Appl., 422(2):422–441, 2007
work page 2007
-
[11]
S. Pierce. Discriminant preserving linear maps.Linear Multilinear Algebra, 8(2):101–114, 1979
work page 1979
-
[12]
S. Pierce et al. A survey of linear preserver problems.Linear Multilinear Algebra, 33(1-2), 1992
work page 1992
-
[13]
M. Radić. A definition of the determinant of a rectangular matrix.Glas. Mat. Ser. III, 1(21):17–22, 1966
work page 1966
-
[14]
M. Radić. A generalization of the determinant of a square matrix and some of its applications in geometry.Mat. (Zagreb), 20(2):19–36, 1991
work page 1991
-
[15]
M. Radić. About a determinant of rectangular2×nmatrix and its geo- metric interpretation.Beitr. Algebra Geom., 46(2):321–349, 2005
work page 2005
-
[16]
M. Radić. Areas of certain polygons in connection with determinants of rectangular matrices.Beitr. Algebra Geom., 49(1):71–96, 2008
work page 2008
-
[17]
P. Šemrl. Linear preserver problems. In L. Hogben, editor,Handbook of linear algebra, Discrete mathematics and its applications, chapter 30-1. CRC press Taylor & Francis, Boca Raton, 2nd ed edition, 2014
work page 2014
-
[18]
Invertibilityoflinearmapspreservingmatrixinvariants
W.C.Waterhouse. Invertibilityoflinearmapspreservingmatrixinvariants. Linear Multilinear Algebra, 13(2):105–113, 1983. 22
work page 1983
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