AN UPPER BOUND FOR THE DETERMINANT OF A
MATRIX WITH GIVEN ENTRY SUM AND SQUARE
SUM
ORTWIN GASPER
HUGO PFOERTNER
MARKUS SIGG
Waltrop, Germany
Munich, Germany
EMail: hugo@pfoertner.org
Freiburg, Germany
EMail: mail@MarkusSigg.de
Upper Bound for the Determinant
of a Matrix
Ortwin Gasper, Hugo Pfoertner
and Markus Sigg
vol. 10, iss. 3, art. 63, 2009
Title Page
Contents
Received:
05 March, 2009
Accepted:
15 September, 2009
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
15A15, 15A45, 26D07.
Key words:
Determinant, Matrix Inequality, Hadamard’s Determinant Theorem, Hadamard
Matrix.
Abstract:
By deducing characterisations of the matrices which have maximal determinant
in the set of matrices with given entry sum and square sum, we prove the inequality | det M | ≤ |α|(β − δ)(n−1)/2 for real n × n-matrices M , where nα and nβ
are the sum of the entries and the sum of the squared entries of M , respectively,
and δ := (α2 − β)/(n − 1), provided that α2 ≥ β. This result is applied to find
an upper bound for the determinant of a matrix whose entries are a permutation
of an arithmetic progression.
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Contents
1
Introduction
3
2
Conventions
4
3
Main Theorem
5
4
Application
13
Upper Bound for the Determinant
of a Matrix
Ortwin Gasper, Hugo Pfoertner
and Markus Sigg
vol. 10, iss. 3, art. 63, 2009
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1.
Introduction
Let n ≥ 2 be a positive integer and a = (a1 , . . . , an2 ) a vector of real numbers. What
is the maximal determinant D(a) of a matrix whose elements are a permutation of
the entries of a? The answer is unknown even for the special case a := (1, . . . , n2 ) if
n > 6, see [4]. By computational optimisation using algorithms like tabu search, we
have found matrices with the following determinants, which thus are lower bounds
for D(1, . . . , n2 ):
n lower bound for D(1, . . . , n2 )
2
10
3
412
4
40 800
5
6 839 492
6
1 865 999 570
7
762 150 368 499
8
440 960 274 696 935
9
346 254 605 664 223 620
10 356 944 784 622 927 045 792
It would be nice to also have a good upper bound for D(1, . . . , n2 ). We will
show how to find an upper bound by treating the problem of determining D(a) as
a continuous optimisation task. This is done by maximising the determinant under
two equality contraints: by fixing the sum and the square sum of the matrix’s entries.
Our result is a characterisation of the matrices with maximal determinant in the
set of matrices with given entry sum and square sum, and a general inequality for
the absolute value of the determinant of a matrix.
For the problem of finding D(1, . . . , n2 ), the upper bound derived in this way
turns out to be quite sharp. So here we have an example where analytical optimisation gives valuable information about a combinatorial optimisation problem.
Upper Bound for the Determinant
of a Matrix
Ortwin Gasper, Hugo Pfoertner
and Markus Sigg
vol. 10, iss. 3, art. 63, 2009
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2.
Conventions
Throughout this article, let n > 1 be a natural number and N := {1, . . . , n}. Matrix
always means a real n × n matrix, the set of which we denote by M.
For M ∈ M and i, j ∈ N we denote by Mi the i-th row of M , by M j the j-th
column of M , and by Mi,j the entry of M at position (i, j). If M is a matrix or a
row or a column of a matrix, then by s(M ) we denote the sum of the entries of M
and by q(M ) the sum of their squares.
The identity matrix is denoted by I. By J we name the matrix which has 1 at all
of its fields, while e is the column vector in Rn with all entries being 1. Matrices of
the structure xI +yJ will play an important role, so we state some of their properties:
Lemma 2.1. Let x, y ∈ R and M := xI + yJ. Then we have:
Upper Bound for the Determinant
of a Matrix
Ortwin Gasper, Hugo Pfoertner
and Markus Sigg
vol. 10, iss. 3, art. 63, 2009
Title Page
1. det M = xn−1 (x + ny)
Contents
2. M is invertible if and only if x 6∈ {0, −ny}.
3. If M is invertible, then M −1 = x1 I −
y
J.
x(x+ny)
Proof. Since J = eeT , it holds that
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M e = (xI + yeeT )e = (x + yeT e)e = (x + ny)e and M v = (xI + yeeT )v = xv
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n
for all v ∈ R with v ⊥ e. Hence M has the eigenvalue x with multiplicity n − 1
and the simple eigenvalue x + ny. This shows (1). (2) is an immediate consequence
of (1). (3) can be verified by a straight calculation.
Close
3.
Main Theorem
Let α, β ∈ R with β > 0 and Mα,β := {M ∈ M : s(M ) = nα, q(M ) = nβ}.
Furthermore, let
α2 − β
δ :=
.
n−1
In the proof of the following lemma, matrices are specified whose determinants
will later turn out to be the greatest possible:
Lemma 3.1.
Upper Bound for the Determinant
of a Matrix
Ortwin Gasper, Hugo Pfoertner
and Markus Sigg
vol. 10, iss. 3, art. 63, 2009
1. Mα,β 6= ∅ if and only if α2 ≤ nβ. If α2 ≤ nβ, then there exists an M ∈ Mα,β
with
n−1
det M = α(β − δ) 2 .
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Contents
n
2
2
2. If α ≤ β, then there exists an M ∈ Mα,β with det M = β .
2
3. There exists an M ∈ Mα,β with det M 6= 0 if and only if α < nβ.
Proof. (1) Suppose Mα,β 6= ∅, say M ∈ Mα,β . Reading M and J as elements of
2
Rn , the Cauchy inequality shows that
!2
n
X
1
Mi,j
α2 = 2
n i,j=1
1
hM, Ji2
n2
n
X
1
2
2
2
≤ 2 kM k2 kJk2 =
Mi,j
= nβ.
n
i,j=1
=
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1
For the other implication suppose α2 ≤ nβ, i. e. β ≥ δ, and set γ := (β − δ) 2
and M := γI + n1 (α − γ)J. Then M ∈ Mα,β , and by Lemma 2.1
n−1
det M = γ n−1 γ + n n1 (α − γ) = γ n−1 α = α(β − δ) 2 .
1 √
3α
2
(2) Let α ≤ β. First suppose α ≥ 0, so γ := 2
− 1 gives γ 2 ≤ 1. Set
β
p


2 0
p
γ
1
−
γ
p
2
p
β−α
pα
A :=
and B := β − 1 − γ 2
γ
0 .
2
− β−α
α
0
0
1
Upper Bound for the Determinant
of a Matrix
Ortwin Gasper, Hugo Pfoertner
and Markus Sigg
vol. 10, iss. 3, art. 63, 2009
3
Then s(A) = 2α, q(A) = 2β, det A = β, s(B) = 3α, q(B) = 3β, det B = β 2 . In
the case of n = 2k with k ∈ N, use k copies of A to build the block matrix


A
..
,
M := 
.
A
which has the required properties. In the case of n = 2k + 1 with k ∈ N, use k − 1
copies of A to build the block matrix


A
..


.
,
M := 


A
B
which again fulfills the requirements.
n
In the case of α < 0, an M 0 ∈ M−α,β with det M 0 = β 2 exists. For even n, the
matrix M := −M 0 ∈ Mα,β has the requested determinant, while for odd n swapping
two rows of −M 0 gives the desired matrix M .
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(3) If α2 < nβ, then the existence of an M ∈ Mα,β with det M 6= 0 is proved by (1)
in the case of α 6= 0 and by (2) in the case of α = 0. For α2 = nβ and M ∈ Mα,β ,
the calculation in (1) shows that hM, Ji = kM k2 kJk2 . However, this equality holds
only if M is a scalar multiple of J, so we have det M = 0 because of det J = 0.
For α2 ≤ β we have given two types of matrices in Lemma 3.1, the first one
n−1
n
having the determinant α(β − δ) 2 , the second one with the determinant β 2 . The
proof of Theorem 3.3 below will use the fact that for α2 < β the determinant of
the first type is strictly smaller than that of the second type. Indeed, the following
stronger statement holds:
Lemma 3.2. Let α2 ≤ nβ. Then |α|(β − δ)
α2 = β.
n−1
2
≤ β with equality if and only if
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n−x n−1
n−1
for
The proof is completed by applying the AM-GM inequality to f (x)1/n :
f (x) =
x
n−x
n−1
n−1 ! n1
vol. 10, iss. 3, art. 63, 2009
n
2
Proof. This is obvious for α = 0, so let α =
6 0. With f (x) := x
x ∈ [0, n] we have
r n−1
n
2
−
|α|(β − δ) 2 β 2 = f αβ .
1
n
Upper Bound for the Determinant
of a Matrix
Ortwin Gasper, Hugo Pfoertner
and Markus Sigg
Contents
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x + (n − 1) n−x
n−1
≤
=1
n
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with equality if and only if x =
n−x
,
n−1
i. e. if and only if x = 1.
If α2 < nβ, then by Lemma 3.1 there exists an M ∈ Mα,β with det M 6= 0,
and, by possibly swapping two rows of M , det M > 0 can be achieved. As Mα,β
is compact, the determinant function assumes a maximum value on Mα,β . The next
theorem, which is essentially due to O. Gasper, shows that this maximum value is
given by the determinants noted in Lemma 3.1:
2
Theorem 3.3. Let α
( < nβ and M ∈ Mα,β with maximal determinant. Then
(1) M M T = βI
if α2 ≤ β:
n
(2) det M = β 2

j

(3) s(Mi ) = s(M ) = α for all i, j ∈ N
if α2 ≥ β: (4) M M T = (β − δ)I + δJ

n−1

(5) det M = |α|(β − δ) 2
Proof. From Lemma 3.1, we know that det M > 0. The matrix M solves an extremum problem with equality contraints


det X −→ max
(P)
(X ∈ M∗ ),
s(X) = nα

q(X) = nβ
Upper Bound for the Determinant
of a Matrix
Ortwin Gasper, Hugo Pfoertner
and Markus Sigg
vol. 10, iss. 3, art. 63, 2009
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where M∗ is the set of invertible matrices. The Lagrange function of (P) is given by
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L(X, λ, µ) = det X − λ(s(X) − nα) − µ(q(X) − nβ),
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so there exist λ, µ ∈ R with dMdi,j L(M, λ, µ) = 0 for all i, j ∈ N . It is well known
that
d
−1
det M
= (det M ) (M T )
dMi,j
i,j
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−1
(see e. g. [1], 10.6), thus we get (det M ) (M T )
(3.1)
− λM − 2µJ = 0, i. e.
(det M )I = λM M T + 2µJM T .
Close
Suppose λ = 0. Then
(det M )n = det(2µJM T ) = det(2µJ) det M = 0 det M = 0
by applying the determinant function to (3.1). This contradicts det M > 0. Hence
λ 6= 0.
(3.2)
T
T
As M M has diagonal elements q(M1 ), . . . , q(Mn ), and JM has diagonal elements s(M1 ), . . . , s(Mn ), we get
n det M = λq(M ) + 2µs(M ) = λnβ + 2µnα
Upper Bound for the Determinant
of a Matrix
Ortwin Gasper, Hugo Pfoertner
and Markus Sigg
vol. 10, iss. 3, art. 63, 2009
by applying the trace function to (3.1), consequently
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det M = λβ + 2µα.
Contents
(3.3)
The symmetry of (det M )I and the symmetry of λM M T in (3.1) show that
µJM T is symmetric as well. As all rows of JM T are identical, namely equal to
(s(M1 ), . . . , s(Mn )), we obtain
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(3.4)
µs(M1 ) = · · · = µs(Mn ).
In the following, we inspect the cases µ = 0 and µ 6= 0 and prove:
(
µ = 0 =⇒ α2 ≤ β ∧ (1) ∧ (2),
(3.5)
µ 6= 0 =⇒ α2 ≥ β ∧ (3) ∧ (4) ∧ (5).
Case µ = 0: Then (3.3) reads det M = λβ, so taking (3.2) into account and dividing
(3.1) by λ gives βI = M M T , i. e. (1). Part (2) follows by applying the determinant
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√ function to (1). Using the Cauchy inequality and the fact that 1
β M is orthogonal and thus an isometry w.r.t. the euclidean norm k · k2 , we get:
!2
n
X
1
(3.6)
α2 = 2
s(Mi )
n
i=1
n
1 X
≤ 2n
s(Mi )2
n
i=1
=
1
1
1
kM ek22 = βkek22 = βn = β.
n
n
n
Case µ 6= 0: Then s(M1 ) = · · · = s(Mn ) by (3.4). The identity
Upper Bound for the Determinant
of a Matrix
Ortwin Gasper, Hugo Pfoertner
and Markus Sigg
vol. 10, iss. 3, art. 63, 2009
Title Page
s(M1 ) + · · · + s(Mn ) = s(M ) = nα
shows that s(Mi ) = α for all i ∈ N . Taking into account that the determinant is
invariant against matrix transposition, this proves (3). Furthermore, JM T = αJ,
and (3.1) becomes
(3.7)
λM M T = (det M )I − 2µαJ,
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hence
1
(det M − 2µα)
λ
for all i ∈ N , and q(M1 ) = · · · = q(Mn ). With
q(Mi ) = (M M T )i,i =
q(M1 ) + · · · + q(Mn ) = q(M ) = nβ,
this shows that
(3.8)
(M M T )i,i = q(Mi ) = β
for all i ∈ N .
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Let i, j ∈ N with i 6= j. Equation (3.7) gives (M M T )i,k = − λ1 2µα for all
k ∈ N \ {i}, and we get
X
(M M T )i,k
β + (n − 1)(M M T )i,j = (M M T )i,i +
k6=i
=
n
X
(M M T )i,k
k=1
=
=
=
n X
n
X
Mi,p Mk,p
k=1 p=1
n
X
Mi,p s(M p )
p=1
n
X
Mi,p α = s(Mi ) α = α2 ,
so
(3.9)
(M M )i,j
α2 − β
= δ.
=
n−1
Equations (3.8) and (3.9) together prove (4). With Lemma 2.1, this yields
(det M )2 = det(M M T ) = (β − δ)n−1 (β − δ + nδ) = α2 (β − δ)n−1 ,
and taking the square root gives (5). Suppose that α2 < β. Then by Lemma 3.1 there
n
exists an M 0 ∈ Mα,β with det M 0 = β 2 , and by Lemma 3.2,
det M = |α|(β − δ)
vol. 10, iss. 3, art. 63, 2009
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p=1
T
Upper Bound for the Determinant
of a Matrix
Ortwin Gasper, Hugo Pfoertner
and Markus Sigg
n−1
2
n
< β 2 = det M 0 ,
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which contradicts the maximality of det M . Hence α2 ≥ β.
We have now proved (3.5) and are ready to deduce the statements of the theorem:
If α2 < β, then (3.5) shows that µ = 0 and thus (1) and (2). If α2 > β, then (3.5)
shows that µ 6= 0 and thus (3), (4) and (5). Finally suppose that α2 = β. Then
δ = 0, hence (1) ⇐⇒ (4) and (2) ⇐⇒ (5). If µ 6= 0, then (3.5) shows (3), (4) and
(5), from which (1) and (2) follow. If µ = 0, then (3.5) shows (1) and (2), from
which (4) and (5) follow. It remains to prove (3) in the case of α2 = β and µ = 0.
To this purpose, look at (3.6) again, where α2 = β means equality in the Cauchy
inequality, which tells us that (s(M1 ), . . . , s(Mn )) is a scalar multiple of e, hence
s(M1 ) = · · · = s(Mn ), and (3) follows as in the case µ 6= 0.
Upper Bound for the Determinant
of a Matrix
Ortwin Gasper, Hugo Pfoertner
and Markus Sigg
vol. 10, iss. 3, art. 63, 2009
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4.
Application
The following is a more application-oriented extract of Theorem 3.3:
Proposition 4.1. Let M ∈ M, α := n1 s(M ), β := n1 q(M ) and δ :=
α2 < β
2
α =β
α2 > β
=⇒
=⇒
=⇒
α2 −β
.
n−1
Then:
n
| det M | ≤ β 2
| det M | ≤ |α|(β − δ)
| det M | ≤ |α|(β − δ)
n−1
2
n−1
2
=β
Upper Bound for the Determinant
of a Matrix
Ortwin Gasper, Hugo Pfoertner
and Markus Sigg
n
2
n
< β2
vol. 10, iss. 3, art. 63, 2009
2
Proof. This is clear if det M = 0. In the case of det M 6= 0, we get α < nβ by
Lemma 3.1, and the stated inequalities are true by Lemma 3.2 and Theorem 3.3.
For M ∈ M with |Mi,j | ≤ 1 for all i, j ∈ N , Proposition 4.1 tells us that
! n2
! n2
n
n
n
n
1 X
1 X 2
≤
= n2,
Mi,j
1
(4.1)
| det M | ≤ β 2 =
n i,j=1
n i,j=1
which is simply the determinant theorem of Hadamard [3]. If Mi,j ∈ {−1, 1} for all
i, j ∈ N and | det M | = nn/2 , i. e. M is a Hadamard matrix, then Proposition 4.1
shows that α2 ≤ β must hold. For a Hadamard matrix M , the value s(M ) is called
the excess of M . Since q(M ) = n2 in the case of Mi,j ∈ {−1, 1}, Proposition 4.1
yields an upper bound for the excess, known as Best’s inequality [2]:
(4.2)
M is a Hadamard matrix
=⇒
3
s(M ) ≤ n 2
The results (4.1) and (4.2), which both can be proved more directly, are mentioned
here just as by-products of Proposition 4.1. In the following, we are interested only
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in the case α2 ≥ β, where the inequality
| det M | ≤ |α|(β − δ)
n−1
2
=: g(M )
n
holds. Note that Lemma 3.2 states that g(M ) < β 2 is true for α2 < β also, but
| det M | is not necessarily bounded by g(M ) in this situation:
1 0
M :=
, | det M | = 1 , g(M ) = 0.
0 −1
We are now going to apply Proposition 4.1 to the problem stated in the introduction. This problem is a special case of finding an upper bound for the determinant of
matrices whose entries are a permutation of an arithmetic progression:
Proposition 4.2. Let p, q be real numbers with q > 0 and M a matrix whose entries
are a permutation of the numbers p, p + q, . . . , p + (n2 − 1)q. Set
r :=
p n2 − 1
+
q
2
and
% :=
n3 + n2 + n + 1
.
12
Then
n
2
| det M | ≤ n q
n
n4 − 1
r +
12
2
Upper Bound for the Determinant
of a Matrix
Ortwin Gasper, Hugo Pfoertner
and Markus Sigg
vol. 10, iss. 3, art. 63, 2009
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n2
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and
2
r >%
=⇒
n n
| det M | ≤ n q |r|%
n−1
2
n
2
<n q
n
n4 − 1
r +
12
2
n2
Close
.
Proof. For α := n1 s(M ) and β := n1 q(M ) a calculation shows that α2 − β = n(n −
1)q 2 (r2 − %), hence (α2 > β ⇐⇒ r2 > %). The bounds noted in Proposition 4.1
yield the asserted inequalities for | det M |.
Corollary 4.3. If M is a matrix whose entries are a permutation of 1, . . . , n2 , then
n2 + 1
| det M | ≤ n
2
n
n3 + n2 + n + 1
12
n−1
2
.
Proof. Apply Proposition 4.2 to (p, q) := (1, 1). For r = (n2 + 1)/2 it is easy to see
that r2 > %, which yields the stated bound.
Comparing the lower bounds for D(1, . . . , n2 ) noted in the introduction with the
upper bounds resulting from rounding down the values given by Corollary 4.3 shows
that the quality of these upper bounds is quite convincing:
Upper Bound for the Determinant
of a Matrix
Ortwin Gasper, Hugo Pfoertner
and Markus Sigg
vol. 10, iss. 3, art. 63, 2009
Title Page
n determinant of best known matrix upper bound given by Corollary 4.3
2
10
11
3
412
450
4
40 800
41 021
5
6 839 492
6 865 625
6
1 865 999 570
1 867 994 210
7
762 150 368 499
762 539 814 814
8
440 960 274 696 935
441 077 015 225 642
9
346 254 605 664 223 620
346 335 386 150 480 625
10
356 944 784 622 927 045 792
357 017 114 947 987 625 629
These are the record matrices R(n) corresponding to the noted determinants:
4
R(2) =
1
2
,
3

9
R(3) = 4
2
3
8
6

5
1 ,
7

12
3
R(4) = 
14
5
13
8
1
11
6
16
9
4

2
7
 ,
10
15
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
25
7

R(5) = 
6
10
16

46
9

39

R(7) = 
26
20

4
32
42
48
11
22
8
28
16
15
36
44
17
40
19
3
15
24
12
13
1
9
14
23
2
18
11
3
20
22
8

4
17

5
,
19
21
2
30
34
41
6
25
37
27
7
13
47
33
38
10
24
14
29
1
23
49
35

18
31

5

21
 ,
45

12
43

68
67

30

56

R(9) = 
23
1

45

15
64

1
 53

 46

 79

 17
R(10) = 
 68

100

 21

 69
52
7
37
20
50
14
38
74
77
52
2
57
63
42
75
93
16
72
15
70
12
43
79
8
54
32
31
35
75
61
3
47
49
87
76
31
29
99
23

36
3

13
R(6) = 
14

20
26

1
44

57

28
R(8) = 
25

19

59
26
62
10
53
27
63
21
80
39
13
84
65
4
71
58
50
35
9
32
96
73
3
49
42
11
65
17
51
57
81
94
45
5
30
85
62
48
44
11
26
61
71
60
18
66
46
16
6
82
20
78
95
6
56
34
73
24
36
24
35
7
22
4
18
29
33
40
81
72
22
5
59
28
39
91
83
51
38
7
8
43
90
55
21
25
34
2
19
9
12
35
2
22
36
60
39
54
58
78
25
4
44
48
24
69
19
54
22
28
10
27
37
64
97
74
92
17
15
16
33
29
1
20
3
51
55
42
33
9
47
52
14
49
4
34
56
37
13

34
36

2

41

70
,
76

47

9
55
41
66
13
77
86
14
67
89
40
12
5
23
10
12
32
30
40
43
23
64
5
46
30
10

60
33

98

26

80
.
19

88

25

18
59

8
11

31
 ,
28

6
27
50
15
11
41
48
6
58
31
53
45
38
18
7
16
21
62

32
61

29

27
,
63

24

8
17
Upper Bound for the Determinant
of a Matrix
Ortwin Gasper, Hugo Pfoertner
and Markus Sigg
vol. 10, iss. 3, art. 63, 2009
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Calculating the matrix M M T for each record matrix M reveals that M M T has
roughly the structure (β − δ)I + δJ that was noted in Theorem 3.3 for the optimal
matrices of the corresponding real optimisation problem.
Upper Bound for the Determinant
of a Matrix
Ortwin Gasper, Hugo Pfoertner
and Markus Sigg
vol. 10, iss. 3, art. 63, 2009
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Contents
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References
[1] D.S. BERNSTEIN, Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory, Princeton University Press, 2005.
[2] M.R. BEST, The excess of a Hadamard matrix. Nederl. Akad. Wet., Proc. Ser. A,
80 (1977), 357–361.
[3] J. HADAMARD, Résolution d’une question relative aux déterminants, Darboux
Bull., (2) XVII (1893), 240–246.
Upper Bound for the Determinant
of a Matrix
Ortwin Gasper, Hugo Pfoertner
and Markus Sigg
vol. 10, iss. 3, art. 63, 2009
[4] N.J.A. SLOANE, The Online Encyclopedia of Integer Sequences, id:A085000.
[ONLINE: http://www.research.att.com/~njas/sequences/
A085000].
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