Matrices with all minors of some fixed order being equal: the rank, dimension and characteristic property
Investigated in this paper is aclass $\mathfrak{M}$ of matrices (over an arbitrary field) in whichall minors of some fixed order $k$ are equal and nonzero. It isestablished that the rank of such matrices equals to $k$. Thepossible values for the dimension of a matrix in $\mathfrak{M}$ arefound. A ne...
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| author | Trebenko, D. Ya. Trebenko, O. O. |
| author_facet | Trebenko, D. Ya. Trebenko, O. O. |
| author_institution_txt_mv | [
{
"author": "D. Ya. Trebenko",
"institution": "Національний педагогічний університет імені М. П. Драгоманова, Київ"
},
{
"author": "O. O. Trebenko",
"institution": null
}
] |
| author_sort | Trebenko, D. Ya. |
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| datestamp_date | 2023-03-02T07:56:20Z |
| description | Investigated in this paper is aclass $\mathfrak{M}$ of matrices (over an arbitrary field) in whichall minors of some fixed order $k$ are equal and nonzero. It isestablished that the rank of such matrices equals to $k$. Thepossible values for the dimension of a matrix in $\mathfrak{M}$ arefound. A necessary and sufficient condition for a matrix to belongto the class $\mathfrak{M}$ is also given. |
| first_indexed | 2026-08-04T01:08:46Z |
| format | Article |
| fulltext |
Збiрник праць Iн-ту математики НАН України 2019, т. 16, № 3, 219–229
D. Ya. Trebenko1, O. O. Trebenko2
1 National Pedagogical Dragomanov University;
trebenko@npu.edu.ua
2 National Pedagogical Dragomanov University;
trebenko@npu.edu.ua
Matrices with all minors of some fixed
order being equal: the rank, dimension
and characteristic property
У статтi дослiджується клас M матриць (над довiльним полем),
в яких всi мiнори деякого фiксованого порядку k – рiвнi i вiдмiннi
вiд 0. Встановлено, що ранг таких матриць дорiвнює k. Знайдено
можливi значення для розмiрностi матрицi з класу M. Дано та-
кож необхiдну i достатню умову для того, щоб матриця належала
до класу M.
Investigated in this paper is a class M of matrices (over an arbitrary
field) in which all minors of some fixed order k are equal and nonzero.
It is established that the rank of such matrices equals to k. The
possible values for the dimension of a matrix in M are found. A
necessary and sufficient condition for a matrix to belong to the class
M is also given.
c© D. Ya. Trebenko, O. O. Trebenko, 2019
220 D. Ya. Trebenko, O. O. Trebenko
1. Introduction
Matrices with all principal minors of some fixed order being equal
were studied by R.C.Thompson in [1] and [2]. In [1] a classification
was obtained for symmetric matrices having all principal minors of
order t equal, for three consecutive values of t less than the rank of
A. A similar result, a classification for real symmetric matrices such
that all principal minors of order t are equal and all nonprincipal
minors are of fixed sign for two consecutive values of t less than the
rank of A, is presented in [2]. The paper [2] also characterizes square
matrices A over an arbitrary field in which the condition on the
principal minors of A is weakened: it is required that all principal
minors of order t are equal for one fixed value of t less then the
rank of A; while the condition on nonprincipal minors of order t is
strengthened: it is required that they are also equal.
Investigated in this paper is a classM of matrices (not only square
and over an arbitrary field) in which all minors of some fixed order k
are equal and nonzero. It is established that the rank of such matrices
equals to k. The possible values for the dimension of a matrix in M
are found. A necessary and sufficient condition for a matrix to belong
to the class M is also given. As an example illustrating main results,
a classification is found for matrices that have all minors of order 2
equal and nonzero.
2. Notation
Let A be a m × n-matrix over an arbitrary field. For A, we use
A>, A∗, rankA, detA to stand for the transpose matrix, the adjoint
matrix, the rank and the determinant of A, respectively.
By Aj we mean j-th column of A (j ∈ 1, n) and Ai is used
to denote i-th row (i ∈ 1,m). In addition, we use the notation
(Aj1Aj2 ...Ajs) for the submatrix formed by selecting from A a subset
of columns Aj1 , Aj2 , ..., Ajs in the same relative position.
Remark that a class M of matrices over a field in which all minors
Matrices with all minors of some fixed order being equal 221
of some fixed order k are equal to w 6= 0 is closed under taking
submatrices and transposes.
3. Main results
Теорема 1. Let P be a field and A be a m × n-matrix over P in
which all minors of order k are equal and nonzero. Then:
(i) rankA = k;
(ii) k ≤ m,n ≤ k + 1.
Доведення. (i) Let all minors of order k of the matrix A be equal to
w. Since, by theorem’s condition, w 6= 0, obviously, rankA ≥ k.
If k = 1 then A = (aij) where aij = w. In the case when w 6= 0
the rank of the matrix A equals to 1 and the assertion of the theorem
is valid.
Let k > 1. Assume that the rank of the matrix A is greater
than k. Then there exist (k+1) linearly independent rows and (k+
1) linearly independent columns in A such that the corresponding
square submatrix B of order k + 1 of the matrix A is nonsingular:
B = (bij), 1 ≤ i ≤ k + 1, 1 ≤ j ≤ k + 1. In this case, for the matrix
B, there exists an inverse matrix B−1:
B−1 = (detB)−1B∗
where B∗ is an adjoint matrix to the matrix B. Since all minors of
order k of the matrix B are equal to w,
B−1=(detB)−1
w −w w ... (−1)k+1w
−w w −w ... (−1)k+2w
w −w w ... (−1)k+3w
... ... ... ... ...
(−1)k+1w (−1)k+2w (−1)k+3w ... (−1)2kw
.
In the case w 6= 0, the rank of the matrix B−1 equals to 1. Since
k > 1, it implies that B−1 is singular, which contradicts to the choice
of B. Hence, the assumption is not valid and rankA = k.
222 D. Ya. Trebenko, O. O. Trebenko
(ii) Let now show that the number of columns (as well as the
number of rows) of the matrix A is equal to k or k + 1. Obviously,
k ≤ m,n.
Let m ≤ n. Assume n ≥ k+2 and consider k× (k+2)-submatrix
C of the matrix A. All minors of order k of the matrix C are equal
to w 6= 0, therefore, in view of (i), rankC = k.
Denote by Cj the j-th column of C.
Since rankC = k and the determinant of the matrix (C1C2...Ck),
obtained by deleting from C both column (k+1) and column (k+2),
is equal to w, we get that the system of vectors C1, C2, ..., Ck
is linearly independent and is the basis of the system of vectors
C1, C2, ..., Ck, Ck+1, Ck+2. Therefore, vector-columns Ck+1 and
Ck+2 can be expressed as the linear combinations of C1, C2, ..., Ck:
Ck+1 =
k∑
i=1
siC
i, Ck+2 =
k∑
i=1
liC
i, si, li ∈ P.
Consider the determinant of the matrix, obtained by deleting
from C both column k and column (k + 2):
det(C1C2...Ck−1Ck+1) = det(C1C2...Ck−1(
k∑
i=1
siC
i)) =
= det(C1C2...Ck−1(skC
k)) = skdet(C
1C2...Ck−1Ck).
Since both det(C1C2...Ck−1Ck+1) and det(C1C2...Ck−1Ck) are mi-
nors of order k of the matrix C, they are equal to w 6= 0, hence,
sk = 1.
Consider now the determinant of the matrix, obtained by deleting
from C both column (k − 1) and column (k + 2):
det(C1C2...Ck−2CkCk+1) = det(C1C2...Ck−2Ck(
k∑
i=1
siC
i)) =
= det(C1C2...Ck−2Ck(sk−1C
k−1)) = sk−1det(C
1C2...Ck−2CkCk−1) =
Matrices with all minors of some fixed order being equal 223
= −sk−1det(C1C2...Ck−2Ck−1Ck).
Since both det(C1C2...Ck−2CkCk+1) and det(C1C2...Ck−2Ck−1Ck)
are minors of order k of the matrix C, they are equal to w 6= 0, hence,
sk−1 = −1.
In a similar way, we get sk−2 = 1, sk−3 = −1, ..., s1 = (−1)k+1.
Then
Ck+1 = (−1)k+1C1 + (−1)kC2 + (−1)k−1C3 + ...− Ck−1 + Ck =
=
k∑
i=1
(−1)k+2−iCi.
Repeating the same considerations for the column Ck+2 of the
matrix C, we get:
Ck+2 = (−1)k+1C1 + (−1)kC2 + (−1)k−1C3 + ...− Ck−1 + Ck =
=
k∑
i=1
(−1)k+2−iCi,
hence, Ck+1 = Ck+2. But then the determinant
det(C3...CkCk+1Ck+2) of order k of the matrix, obtained by
deleting from C both column 1 and column 2, is necessarily equal
to 0, which contradicts the condition of the theorem. Therefore,
assumption is not valid and n ≤ k + 1.
It remains to consider the case n ≤ m. Since all minors of order
k of the transpose matrix A> are also equal to w 6= 0, applying the
proven result to A> gives us that the number of columns of A> does
not exceed k + 1, hence, m ≤ k + 1.
The theorem is proven. �
Наслiдок 2. Let A be a k × (k + 1)-matrix over the field P . All
minors of order k of the matrix A are equal and nonzero iff the
following conditions 1)-2) hold:
224 D. Ya. Trebenko, O. O. Trebenko
1) rankA = k;
2) (k + 1)-th column Ak+1 of the matrix A is expressed as the
linear combination:
Ak+1 =
k∑
j=1
(−1)k+2−jAj =
= (−1)k+1A1 + (−1)kA2 + (−1)k−1A3 + ...−Ak−1 +Ak
where Aj is a j-th column of the matrix A, 1 ≤ j ≤ k.
Доведення. Necessity immediately follows from the proof of
Theorem.
Sufficiency. Let the conditions 1)-2) hold for the matrix A and
det(A1A2...Ak) = w 6= 0. Let M be an arbitrary minor of order k of
the matrix A. Then M is a determinant of a matrix, obtained by
deleting from A some column Aj , j ∈ 1, k + 1.
If j = k+ 1 then M = det(A1A2...Ak) = w. Let 1 ≤ j ≤ k. Then
M = det(A1...Aj−1Aj+1...AkAk+1) =
= det(A1...Aj−1Aj+1...Ak(
k∑
j=1
(−1)k+2−jAj)) =
= det(A1...Aj−1Aj+1...Ak((−1)k+2−jAj)) =
= (−1)k+2−jdet(A1...Aj−1Aj+1...AkAj) =
= (−1)k+2−j(−1)k−jdet(A1...Aj−1AjAj+1...Ak) = (−1)2(k+1−j)w = w.
The corollary is proven. �
The next proposition follows immediately from Corollary 1, in
view of the fact that the class M of matrices with all minors of some
fixed order k being equal and nonzero is closed by taking inverse
matrices and submatrices.
Наслiдок 3. Let A be a (k + 1)× (k + 1)-matrix over the field P .
All minors of order k of the matrix A are equal and nonzero iff the
following conditions 1)-2) hold:
Matrices with all minors of some fixed order being equal 225
1) rankA = k;
2) (k + 1)-th column Ak+1 of the matrix A is expressed as the
linear combination:
Ak+1 =
k∑
j=1
(−1)k+2−jAj =
= (−1)k+1A1 + (−1)kA2 + (−1)k−1A3 + ...−Ak−1 +Ak
where Aj is a j-th column of the matrix A, 1 ≤ j ≤ k.
3) (k + 1)-th row Ak+1 of the matrix A is expressed as the linear
combination:
Ak+1 =
k∑
i=1
(−1)k+2−iAi =
= (−1)k+1A1 + (−1)kA2 + (−1)k−1A3 + ...−Ak−1 +Ak
where Ai is a i-th row of the matrix A, 1 ≤ i ≤ k.
Зауваження 1. For arbitrary given field P , w ∈ P\{0} and posi-
tive integer k, there exist matrices over P of the dimensions k × k,
k× (k+1), (k+1)× k, (k+1)× (k+1) having all minors of order
k equal to w. Indeed, one can always indicate a square matrix B of
order k, which determinant is equal to w, e.g.,
A =
w 0 0 ... 0
0 1 0 ... 0
0 0 1 ... 0
... ... ... ... ...
0 0 0 ... 1
.
Consider a k × (k + 1)-matrix A such that B is a submatrix of A
obtained by deleting (k + 1)-th row: B = (A1A2...Ak), and
Ak+1 =
k∑
j=1
(−1)k+2−jAj =
226 D. Ya. Trebenko, O. O. Trebenko
= (−1)k+1A1 + (−1)kA2 + (−1)k−1A3 + ...−Ak−1 +Ak
In view of Corollary 1, all minors of order k of the matrix A are equal
to w. In a similar way, one can construct matrices the dimensions
(k + 1)× k, (k + 1)× (k + 1).
As an illustration to the Theorem, consider the following example
classifying matrices in which all minors of order 2 are equal to some
fixed w 6= 0.
Example 1. Let A be a matrix over a field P . All minors of order
2 of A are equal to w 6= 0 iff A is a matrix of one of the following
types:
1) A =
(
a1 a2
−wa−12 0
)
where a1, a2 ∈ P , a2 6= 0;
2) A =
(
(a2a3 + w)a−14 a2
a3 a4
)
where a2, a3, a4 ∈ P , a4 6= 0;
3) A =
(
a1 a1 + a2 a2
−wa−12 −wa−12 0
)
where a1, a2 ∈ P , a2 6= 0;
4) A =
(
(a2a3 + w)a−14 (a2a3 + w)a−14 + a2 a2
a3 a3 + a4 a4
)
where a2, a3, a4 ∈ P , a4 6= 0;
5) A =
a1 −wa−12
a1 + a2 −wa−12
a2 0
where a1, a2 ∈ P , a2 6= 0;
6) A =
(a2a3 + w)a−14 a3
(a2a3 + w)a−14 + a2 a3 + a4
a2 a4
where a2, a3, a4 ∈ P ,
a4 6= 0;
Matrices with all minors of some fixed order being equal 227
7) A =
a1 a1 + a2 a2
a1 − wa−12 a1 + a2 − wa−12 a2
−wa−12 −wa−12 0
where a1, a2 ∈ P ,
a2 6= 0;
8) A =
(a2a3 + w)a−14 (a2a3 + w)a−14 + a2 a2
(a2a3 + w)a−14 + a3 (a2a3 + w)a−14 + a2 + a3 + a4 a2 + a4
a3 a3 + a4 a4
where a2, a3, a4 ∈ P , a4 6= 0.
Indeed, by Theorem, rankA = 2, the number of rows and columns
is 2 or 3.
Case 1. Let A be a square matrix of order 2: A =
(
a1 a2
a3 a4
)
,
a1a4 − a2a3 = w 6= 0, a1, a2, a3, a4 ∈ P . If a4 = 0 then a2a3 =
−w. Since w 6= 0, we have a2 6= 0 and a3 = −wa−12 , hence, A =(
a1 a2
−wa−12 0
)
and A is of type 1). If a4 6= 0 then a1 = (a2a3 +
w)a−14 , hence,
A =
(
(a2a3 + w)a−14 a2
a3 a4
)
and A is of type 2).
Case 2. Let A be a 2 × 3-matrix. Then, by Corollary, its
2-nd column is a sum of the 1-st and 3-rd columns: A =(
a1 a1 + a2 a2
a3 a3 + a4 a4
)
where a1a4 − a2a3 = w, a1, a2, a3, a4 ∈ P .
If a4 = 0 then a2 6= 0 and a3 = −wa−12 , hence,
A =
(
a1 a1 + a2 a2
−wa−12 −wa−12 0
)
and A is of type 3). If a4 6= 0 then a1 = (a2a3 + w)a−14 , hence,
A =
(
(a2a3 + w)a−14 (a2a3 + w)a−14 + a2 a2
a3 a3 + a4 a4
)
228 D. Ya. Trebenko, O. O. Trebenko
and A is of type 4).
Case 3. Let A be a 3× 2-matrix. Then the transpose matrix A>
is a matrix of type 3) or type 4), hence, A is a matrix of type 5) or
type 6).
Case 4. Let A be a 3 × 3-matrix. Then, by Corollary 2,
its 2-nd column is a sum of the 1-st and 3-rd columns, whi-
le its 2-nd row is a sum of the 1-st and 3-rd rows: A = a1 a1 + a2 a2
a1 + a3 a1 + a2 + a3 + a4 a2 + a4
a3 a3 + a4 a4
where a1, a2, a3, a4 ∈ P ,
a1a4 − a2a3 = w 6= 0. If a4 = 0 then a2 6= 0, a3 = −wa−12 ,
hence, A =
a1 a1 + a2 a2
a1 − wa−12 a1 + a2 − wa−12 a2
−wa−12 −wa−12 0
and A is a
matrix of type 7). If a4 6= 0 then a1 = (a2a3 + w)a−14 , hence, A = (a2a3 + w)a−14 (a2a3 + w)a−14 + a2 a2
(a2a3 + w)a−14 + a3 (a2a3 + w)a−14 + a2 + a3 + a4 a2 + a4
a3 a3 + a4 a4
and A is of type 8).
Corollaries 1,2 and direct calculations show that the matrices of
types 1)-8) have all minors of order 2 equal to w.
4. Conclusion
In this paper, we have established that the rank of a matrix having
all minors of order k equal and nonzero is equal to k. The number
of columns of such matrices is k or k + 1 (as well as the number of
rows). Using the necessary and sufficient condition for a matrix to
have all minors of order k equal and nonzero, one can easily classify
all matrices for fixed values of k. In this study, such classification is
given for k = 2.
Matrices with all minors of some fixed order being equal 229
Лiтература
[1] Thompson, R. C. Principal submatrices V: Some results co ncerning
principal submatri ces of arbitrary matrices // Journal of Research of
the National Bureau of Standards. — 1968. – Vol. 72B (Math . Sci.),
No. 2 — pp. 115-125.
[2] Thompson, R. C. Principal submatrices VII: Further results concerni-
ng matrices with equal principal minors // Journal of Research of the
National Bureau of Standards. — 1968. – Vol. 72B (Math . Sci.), No.
4 — pp. 249-252.
|
| id | oai:trim.imath.kiev.ua:article-519 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-08-04T01:08:46Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/7f/fbc36b091e57a8a08fbde0768423157f.pdf |
| spelling | oai:trim.imath.kiev.ua:article-5192023-03-02T07:56:20Z Matrices with all minors of some fixed order being equal: the rank, dimension and characteristic property Trebenko, D. Ya. Trebenko, O. O. Investigated in this paper is aclass $\mathfrak{M}$ of matrices (over an arbitrary field) in whichall minors of some fixed order $k$ are equal and nonzero. It isestablished that the rank of such matrices equals to $k$. Thepossible values for the dimension of a matrix in $\mathfrak{M}$ arefound. A necessary and sufficient condition for a matrix to belongto the class $\mathfrak{M}$ is also given. У статті досліджується клас$\mathfrak{M}$ матриць (над довільним полем), в яких всі міноридеякого фіксованого порядку $k$ -- рівні і відмінні від 0.Встановлено, що ранг таких матриць дорівнює $k$. Знайдено можливізначення для розмірності матриці з класу $\mathfrak{M}$. Дано такожнеобхідну і достатню умову для того, щоб матриця належала до класу$\mathfrak{M}$. Інститут математики НАН України 2019-12-30 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/519 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 16 No. 3 (2019): Фрактальний аналіз та суміжні питання; 219-229 Сборник Трудов Института математики НАН Украины; Том 16 № 3 (2019): Фрактальний аналіз та суміжні питання; 219-229 Збірник Праць Інституту математики НАН України; Том 16 № 3 (2019): Фрактальний аналіз та суміжні питання; 219-229 3083-7529 1815-2910 uk https://trim.imath.kiev.ua/index.php/trim/article/view/519/491 Авторське право (c) 2019 D. Ya. Trebenko, O. O. Trebenko http://creativecommons.org/licenses/by/4.0 |
| spellingShingle | Trebenko, D. Ya. Trebenko, O. O. Matrices with all minors of some fixed order being equal: the rank, dimension and characteristic property |
| title | Matrices with all minors of some fixed order being equal: the rank, dimension and characteristic property |
| title_full | Matrices with all minors of some fixed order being equal: the rank, dimension and characteristic property |
| title_fullStr | Matrices with all minors of some fixed order being equal: the rank, dimension and characteristic property |
| title_full_unstemmed | Matrices with all minors of some fixed order being equal: the rank, dimension and characteristic property |
| title_short | Matrices with all minors of some fixed order being equal: the rank, dimension and characteristic property |
| title_sort | matrices with all minors of some fixed order being equal: the rank, dimension and characteristic property |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/519 |
| work_keys_str_mv | AT trebenkodya matriceswithallminorsofsomefixedorderbeingequaltherankdimensionandcharacteristicproperty AT trebenkooo matriceswithallminorsofsomefixedorderbeingequaltherankdimensionandcharacteristicproperty |