1. Hadamard
Matrices:
Truth &
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Hadamard Matrices:
Truth & Consequences
Raymond Nguyen
Advisor: Peter Casazza
The University of Missouri
Math 8190 (Master’s Project)
Spring 2020
1 / 55
2. Hadamard
Matrices:
Truth &
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Outline
1 Basic Theory of Hadamard Matrices
2 Hadamard Matrix Constructions
3 Applications of Hadamard Matrices
2 / 55
3. Hadamard
Matrices:
Truth &
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Definition & Examples
Properties
The Hadamard Conjecture
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Table of Contents
1 Basic Theory of Hadamard Matrices
Definition & Examples
Properties
The Hadamard Conjecture
2 Hadamard Matrix Constructions
3 Applications of Hadamard Matrices
3 / 55
4. Hadamard
Matrices:
Truth &
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Definition & Examples
Properties
The Hadamard Conjecture
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
What is a Hadamard Matrix?
Definition (Hadamard Matrix)
A square matrix H of order n whose entries are +1 or −1 is called a
Hadamard matrix of order n provided its rows are pairwise orthogonal –
i.e.,
HHT
= n · In. (1)
Note that (1) implies that H has an inverse 1
n
HT
. Consequently, its
columns are also pairwise orthogonal – i.e.,
HT
H = n · In. (2)
4 / 55
6. Hadamard
Matrices:
Truth &
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Definition & Examples
Properties
The Hadamard Conjecture
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
“Anallagmatic Pavement” – J.J. Sylvester (1867)
Hadamard Matrices of Order 1, 2, 4, and 8:
, , , and
6 / 55
7. Hadamard
Matrices:
Truth &
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Definition & Examples
Properties
The Hadamard Conjecture
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
James Joseph Sylvester (1814 – 1897)
(Source: http://mathshistory.st-andrews.ac.uk/)
1. Discovered Hadamard matrices
in 1867.
2. Coined the terms ”matrix”,
”determinant”, and
”discriminant”.
3. Served as the very first math
professor at Johns Hopkins
University (1876-1883).
7 / 55
8. Hadamard
Matrices:
Truth &
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Definition & Examples
Properties
The Hadamard Conjecture
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
The Key Property of a Hadamard Matrix
Recall that the defining property of a Hadamard matrix is that
its rows are orthogonal – i.e., HHT = n · In. This property does not change if
we:
1. Take the transpose of H.
2. Permute the rows of H.
3. Permute the columns of H.
4. Multiply any row by -1.
5. Multiply any column by -1.
Definition (Hadamard Equivalence)
Two Hadamard matrices are called equivalent if they differ by some sequence of
the 5 operations listed above.
8 / 55
9. Hadamard
Matrices:
Truth &
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Definition & Examples
Properties
The Hadamard Conjecture
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
The Normal Form of a Hadamard Matrix
Definition
A Hadamard matrix is called normalized if its first row and first column
consist entirely of +1’s.
For example, here is a normalized Hadamard matrix of order 4:
+ + + +
+ + − −
+ − + −
+ − − +
9 / 55
10. Hadamard
Matrices:
Truth &
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Definition & Examples
Properties
The Hadamard Conjecture
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Normalization
Fact
Every Hadamard matrix is equivalent to a normalized Hadamard matrix.
Given a Hadamard matrix, we can write it in normal form by negating every row
and every column whose first element is −1.
+ − + +
+ − − −
− − − +
+ + − +
negate 3rd row
−
−
−
−
−
−
−
−
−
→
+ − + +
+ − − −
+ + + −
+ + − +
negate 2nd col
−
−
−
−
−
−
−
−
→
+ + + +
+ + − −
+ − + −
+ − − +
10 / 55
11. Hadamard
Matrices:
Truth &
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Definition & Examples
Properties
The Hadamard Conjecture
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Properties of Normalized Hadamard Matrices
Lemma
Suppose H is a normalized Hadamard matrix. With the exception of the first row
(column), every row (column) of H has the following properties:
1) Exactly half of the elements are +1’s and exactly half are −1’s.
2) Exactly half of the +1’s overlap with a +1 in any other given row (column).
+ + + +
+
+
+
(1)
−
−
→
+ + + +
+ + − −
+
+
(1) & (2)
−
−
−
−
−
→
+ + + +
+ + − −
+ − + −
+
(1) & (2)
−
−
−
−
−
→
+ + + +
+ + − −
+ − + −
+ − − +
Up to equivalence, the matrix on the right is the unique Hadamard matrix of order 4.
11 / 55
12. Hadamard
Matrices:
Truth &
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Definition & Examples
Properties
The Hadamard Conjecture
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
How Many Distinct Hadamard Matrices of a Given
Order Exist?
Up to equivalence, there is a unique Hadamard matrix of order m
for m = 1, 2, 4, 8, and 12. For m > 12, the number of distinct Hadamard
matrices can be very large.
order 1 2 4 8 12 16 20 24 28 32
# 1 1 1 1 1 4 3 36 294 > 1 million
(Source: I.M. Wanless, 2005)
12 / 55
13. Hadamard
Matrices:
Truth &
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Definition & Examples
Properties
The Hadamard Conjecture
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
A Necessary Condition for Existence
Lemma
Suppose H is a normalized Hadamard matrix. With the exception of the first row
(column), every row (column) of H has the following properties:
1) Exactly half of the elements are +1’s and exactly half are −1’s.
2) Exactly half of the +1’s overlap with a +1 in any other given row (column).
Theorem
If a Hadamard matrix of order m exists, then m = 1, 2, or a multiple of 4.
[+] ,
+ +
+ −
, or
+ + + + + · · · + + + + + +
+ + · · · · · · + + − − · · · · · · − −
+ · · · + − · · · −
.
.
.
+
13 / 55
14. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Definition Examples
Properties
The Hadamard Conjecture
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Question: Does there exist a Hadamard matrix for
every order that is a multiple of 4?
Answer: Nobody knows ... but go try this at home!
The Hadamard Conjecture
A Hadamard matrix of order 4n exists for all positive integers n.
Raymond Paley (1933): “It seems probable that, whenever m is divisible by 4, it
is possible to construct an orthogonal matrix of order m composed of ±1, but
the general theorem has every appearance of difficulty.”
14 / 55
15. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Definition Examples
Properties
The Hadamard Conjecture
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Jacques Hadamard (1865 – 1963)
(Source: http://mathshistory.st-andrews.ac.uk/)
1. Showed in 1893 that if M is a square
matrix of order n whose entries are ±1,
then det(M) ≤ (
√
n)n
and that this bound
is achieved iff M is a Hadamard matrix.
(This is a special case of Hadamard’s
maximal determinant problem.)
2. Proved that if a Hadamard matrix exists
(i.e., if the bound above is achieved), then
its order must be 1, 2, or a multiple of 4.
3. Was the first to construct Hadamard
matrices of orders 12 and 20.
15 / 55
16. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Sylvester’s Construction
Paley’s Construction
Williamson’s Method
Applications of
Hadamard
Matrices
Table of Contents
1 Basic Theory of Hadamard Matrices
2 Hadamard Matrix Constructions
Sylvester’s Construction
Paley’s Construction
Williamson’s Method
3 Applications of Hadamard Matrices
16 / 55
17. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Sylvester’s Construction
Paley’s Construction
Williamson’s Method
Applications of
Hadamard
Matrices
Kronecker Product
Definition
If A = (aij ) is an m × p matrix and B = (bij ) is a n × q matrix, then the Kronecker
product (or tensor product) A ⊗ B is the mn × pq matrix given by
A ⊗ B =
a11B a12B · · · a1pB
a21B a22B · · · a2pB
.
.
.
am1B am2B · · · ampB
Properties
1) Transpose Property: (A ⊗ B)T
= AT
⊗ BT
.
2) Mixed-Product Property: (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD)
(assuming A, B, C, and D are of the correct size for matrix multiplication.)
17 / 55
18. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Sylvester’s Construction
Paley’s Construction
Williamson’s Method
Applications of
Hadamard
Matrices
Sylvester’s Construction, Part 1
Theorem (Sylvester)
If H1 is a Hadamard matrix of order m and H2 is a Hadamard matrix of
order n then H1 ⊗ H2 is a Hadamard matrix of order mn.
Proof:
(H1 ⊗ H2)(H1 ⊗ H2)T
= (H1 ⊗ H2)(HT
1 ⊗ HT
2 ) (Transpose Prop.)
= (H1HT
1 ) ⊗ (H2HT
2 ) (Mixed-Product Prop.)
= (m · Im) ⊗ (n · In) (Hadamard Matrices)
= mn · Imn.
18 / 55
19. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Sylvester’s Construction
Paley’s Construction
Williamson’s Method
Applications of
Hadamard
Matrices
Sylvester’s Construction, Part 2
Corollary (Sylvester)
There exists a Hadamard matrix of order 2n
for every positive integer n.
Proof: We already know that there exists a Hadamard matrix of order 2:
H =
+ +
+ −
.
Now repeatedly take tensor powers of H to obtain a Hadamard matrix of
order 2n
for any positive integer n.
19 / 55
21. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Sylvester’s Construction
Paley’s Construction
Williamson’s Method
Applications of
Hadamard
Matrices
What We Know So Far
The Hadamard Conjecture
A Hadamard matrix of order 4n exists for all positive integers n.
Multiples of 4 that are less than or equal to 100:
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48,
52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100
Sylvester’s construction [m = 2k
or m = m1m2 where m1 and m2 are Hadamard orders]
21 / 55
22. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Sylvester’s Construction
Paley’s Construction
Williamson’s Method
Applications of
Hadamard
Matrices
Prime Time
Theorem (Umberto Scarpis, 1898)
Suppose p is a prime number. Then we have the following:
1) If p ≡ 3 (mod 4), then there is a Hadamard matrix of order p + 1.
2) If p ≡ 1 (mod 4), then there is a Hadamard matrix of order 2(p + 1).
In 1933, Paley discovered two constructions which generalize the work of Scarpis.
Theorem (Paley, 1933)
Suppose p is a prime number and α 0. Then we have the following:
1) If pα ≡ 3 (mod 4), then there is a Hadamard matrix of order pα + 1.
2) If pα ≡ 1 (mod 4), then there is a Hadamard matrix of order 2(pα + 1).
22 / 55
23. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Sylvester’s Construction
Paley’s Construction
Williamson’s Method
Applications of
Hadamard
Matrices
Paley’s Construction, Type 1
Paley’s 1st Theorem
Suppose p is a prime number and α 0. Then we have the following:
If pα ≡ 3 (mod 4), then there is a Hadamard matrix of order m = pα + 1.
Sketch of Proof: Label the elements of GF(pα) as a0, a1, a2, . . . in some order.
Let Q = (qij ) be a matrix of order pα whose entries are given by qij = χ(ai − aj ) where χ is the
quadratic character on GF(pα). That is,
χ(b) =
0, if b = 0
+1, if b is a non-zero perfect square in GF(pα)
−1, if b is not a perfect square in GF(pα)
.
Form the m × m matrix C =
h
0 1T
−1 Q
i
and write H = Im + C.
Then H is a Hadamard matrix of order m = pα + 1.
[Note: The key to the proof is that C is an anti-symmetric conference matrix.
That is, C = −CT (anti-symmetric). Also, C has all 0’s along the diagonal, ±1 elsewhere, and
CCT = (m − 1)Im (conference matrix).]
Anti-symmetry: qji = χ(aj − ai ) = χ((−1)(ai − aj )) = χ(−1)χ(ai − aj ) = (−1)χ(ai − aj ) = −qij .
23 / 55
24. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Sylvester’s Construction
Paley’s Construction
Williamson’s Method
Applications of
Hadamard
Matrices
Example Using Paley’s 1st Theorem
Let’s construct a Hadamard matrix of order 12 (= 11 + 1).
Here we have pα
= 11 ≡ 3 (mod 4) and pα
+ 1 = 12 so such a matrix must exist.
Write the elements of GF(11) as a0 = 0, a1 = 1, a2 = 2, . . . .
Observe that the perfect squares of GF(11) are 1, 3, 4, 5, and 9.
That is, 12
≡ 1, 52
≡ 3, 22
≡ 4, 42
≡ 5, and 32
≡ 9.
Then the quadratic character matrix Q and the Hadamard matrix H are:
Q =
0 − + − − − + + + − +
+ 0 − + − − − + + + −
− + 0 − + − − − + + +
+ − + 0 − + − − − + +
+ + − + 0 − + − − − +
+ + + − + 0 − + − − −
− + + + − + 0 − + − −
− − + + + − + 0 − + −
− − − + + + − + 0 − +
+ − − − + + + − + 0 −
− + − − − + + + − + 0
−→ H =
+ + + + + + + + + + + +
− + − + − − − + + + − +
− + + − + − − − + + + −
− − + + − + − − − + + +
− + − + + − + − − − + +
− + + − + + − + − − − +
− + + + − + + − + − − −
− − + + + − + + − + − −
− − − + + + − + + − + −
− − − − + + + − + + − +
− + − − − + + + − + + −
− − + − − − + + + − + +
.
Observe that Q is a circulant matrix. That is, each row is obtained from the row above
it by a cyclic permutation.
24 / 55
25. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Sylvester’s Construction
Paley’s Construction
Williamson’s Method
Applications of
Hadamard
Matrices
Paley’s Construction, Type 2
Paley’s 2nd Theorem
Suppose p is a prime number and α 0. Then we have the following:
If pα ≡ 1 (mod 4), then there is a Hadamard matrix of order m = 2(pα + 1).
Sketch of Proof: As in Paley’s first theorem, let Q = (qij ) be a matrix of order pα whose entries are
given by qij = χ(ai − aj ).
Let n = pα + 1. Form the n × n matrix C =
0 1T
1 Q
and this time write H = C ⊗
+ +
+ −
+ In ⊗
+ −
− −
.
Then H is a Hadamard matrix of order m = 2n = 2(pα + 1).
[Note: The key to the proof is that C is a symmetric conference matrix.
That is, C = CT (symmetric). Also, C has all 0’s along the diagonal, ±1 elsewhere, and
CCT = (n − 1)In (conference matrix).]
Symmetry: qji = χ(aj − ai ) = χ((−1)(ai − aj )) = χ(−1)χ(ai − aj ) = (+1)χ(ai − aj ) = qij .
25 / 55
26. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Sylvester’s Construction
Paley’s Construction
Williamson’s Method
Applications of
Hadamard
Matrices
Example using Paley’s 2nd Theorem
Let’s construct another Hadamard matrix of order 12 [= 2(5 + 1)] . . . this time using Paley’s second
theorem.
Here we have pα = 5 ≡ 1 (mod 4) and 2(pα + 1) = 12 so such a matrix must exist.
Write the elements of GF(5) as a0 = 0, a1 = 1, a2 = 2, . . . .
Observe that the perfect squares of GF(5) are 1 and 4.
That is, 12 ≡ 1 and 22 ≡ 4.
Then the quadratic character matrix Q, the conference matrix C, and the Hadamard matrix H are:
Q =
0 + − − +
+ 0 + − −
− + 0 + −
− − + 0 +
+ − − + 0
#
−→ C =
0 + + + + +
+ 0 + − − +
+ + 0 + − −
+ − + 0 + −
+ − − + 0 +
+ + − − + 0
−→ H =
+ − + + + + + + + + + +
− − + − + − + − + − + −
+ + + − + + − − − − + +
+ − − − + − − + − + + −
+ + + + + − + + − − − −
+ − + − − − + − − + − +
+ + − − + + + − + + − −
+ − − + + − − − + − − +
+ + − − − − + + + − + +
+ − − + − + + − − − + −
+ + + + − − − − + + + −
+ − + − − + − + + − − −
.
Recall: H = C ⊗
+ +
+ −
+ In ⊗
+ −
− −
.
26 / 55
27. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Sylvester’s Construction
Paley’s Construction
Williamson’s Method
Applications of
Hadamard
Matrices
Raymond E.A.C. Paley (January 1907 - April 1933)
(Source: http://mathshistory.st-andrews.ac.uk/)
[Credit: The Times (Canada)]
27 / 55
28. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Sylvester’s Construction
Paley’s Construction
Williamson’s Method
Applications of
Hadamard
Matrices
What We Know So Far
The Hadamard Conjecture
A Hadamard matrix of order 4n exists for all positive integers n.
Multiples of 4 that are less than or equal to 100:
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48,
52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100
Sylvester’s construction [m = 2k
or m = m1m2 where m1 and m2 are Hadamard orders]
Paley’s construction, type 1 [m = pα
+ 1 where pα
≡ 3 (mod 4)]
Paley’s construction, type 2 [m = 2(pα
+ 1) where pα
≡ 1 (mod 4)]
Sylvester/Paley type 1 [m = m1m2 where m1 is Sylvester and m2 is Paley-type-1]
28 / 55
29. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Sylvester’s Construction
Paley’s Construction
Williamson’s Method
Applications of
Hadamard
Matrices
A Path Forward
Theorem (John Williamson, 1944)
If A, B, C and D are square (1,-1) matrices of order n which satisfy
1. AAT + BBT + CCT + DDT = 4n · In and
2. XY T = YXT
for any two distinct matrices X, Y ∈ {A, B, C, D},
then H =
A B C D
−B A −D C
−C D A −B
−D −C B A
is a Hadamard matrix of order 4n.
Proof: Let’s consider M = HHT as a block matrix with n × n blocks. Observe:
1. Mii = AAT + BBT + CCT + DDT (1)
= 4n · In.
2. M1,2 = −ABT + BAT − CDT + DCT (2)
= 0 + 0 = 0. (similarly for other i 6= j)
Therefore, we have M = HHT =
4n·In 0 0 0
0 4n·In 0 0
0 0 4n·In 0
0 0 0 4n·In
= 4n · I4n.
29 / 55
30. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Sylvester’s Construction
Paley’s Construction
Williamson’s Method
Applications of
Hadamard
Matrices
Williamson’s Method
Theorem (John Williamson, 1944)
If A, B, C, and D are square matrices of order n whose entries are ±1 which satisfy
1. AAT + BBT + CCT + DDT = 4n · In and
2. XY T = YXT
for any two distinct matrices X, Y ∈ {A, B, C, D},
then H =
A B C D
−B A −D C
−C D A −B
−D −C B A
is a Hadamard matrix of order 4n.
To further narrow the search for Hadamard matrices, Williamson made 2 simplifying assumptions:
1. A, B, C, and D are symmetric. (This reduces the second condition XY T = YXT to saying that
A, B, C, and D commute.)
2. A, B, C, and D are circulant. (This guarantees that A, B, C, and D commute. Thus, if we
restrict A, B, C, and D to symmetric, circulant matrices then we need only check that the first
condition is satisfied.)
30 / 55
31. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Sylvester’s Construction
Paley’s Construction
Williamson’s Method
Applications of
Hadamard
Matrices
Breakthroughs Using Williamson’s Method
Using Williamson’s method, many previously unknown Hadamard matrices have
been discovered for various orders. We call these Williamson matrices.
1. Williamson (1944) used number theory to find Williamson matrices of order
148 (= 4 · 37) and 172 (= 4 · 43).
2. Baumert, Golomb, and Hall (1962) used a computer to find a Williamson
matrix of order 92 (= 4 · 23). (This is the missing order from three slides
ago.)
3. Turyn (1972) found an infinite class of Williamson matrices. That is,
if n is odd and 2n − 1 is a prime power, then there is a Williamson matrix
of order 4n.
31 / 55
32. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Sylvester’s Construction
Paley’s Construction
Williamson’s Method
Applications of
Hadamard
Matrices
Hmm ... That’s Odd!
Notice that a lot of effort has gone into finding Hadamard matrices of order 4n where n
is odd. Here’s why:
Theorem
Suppose that for every odd n 0, there exists a Hadamard matrix of order 4n.
Then there exists a Hadamard matrix of order m = 4n for every n 0. (I.e., then the
Hadamard conjecture is true.)
Proof: Let n 0. Write m = 4n = 2i
j for some i ≥ 2 and j odd.
1. If i = 2 then m = 4j where j is odd. By assumption, there exists a Hadamard
matrix of order m = 4j.
2. If i 2 then m = 2k
4j for some k 0. By Sylvester’s corollary, there exists a
Hadamard matrix H1 of order 2k
and by assumption, there exists a Hadamard
matrix H2 of order 4j. Thus, by Sylvester’s theorem, H = H1 ⊗ H2 is a
Hadamard matrix of order m = 2k
4j.
32 / 55
33. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Sylvester’s Construction
Paley’s Construction
Williamson’s Method
Applications of
Hadamard
Matrices
Question: Do Williamson Matrices of Order 4n Exist
for Every Odd Positive Integer n?
Answer
: If the answer is “yes”, then the Hadamard conjecture is true.
Unfortunately, the answer is “no”!
1. Dokovic (1993) showed that there are no Williamson matrices of order
4n for n = 35.
2. Holzmann et. al (2007) showed that there are no Williamson matrices
of order 47, 53, or 59.
3. Therefore, it is not possible to prove the Hadamard conjecture solely
by using Williamson matrices.
33 / 55
34. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Coding Theory
Quantum Computing
Much, Much More
Table of Contents
1 Basic Theory of Hadamard Matrices
2 Hadamard Matrix Constructions
3 Applications of Hadamard Matrices
Coding Theory
Quantum Computing
Much, Much More
34 / 55
35. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Coding Theory
Quantum Computing
Much, Much More
Mariner 9 (May 30, 1971 – October 27, 1972)
(Credit: NASA)
1. Mariner 9 became the first spacecraft to orbit another planet (Mars).
2. It mapped over 85% of the Martian surface and sent back over 7,000 pictures
including images of Olympus Mons, Valles Marineris, and Phobos Deimos.
3. It revolutionized our perception of Mars from a cold, monotone planet to one full
of geologic activity in the past and atmospheric dynamics in the present.
35 / 55
36. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Coding Theory
Quantum Computing
Much, Much More
Olympus Mons (1972)
(Credit: NASA)
1. Olympus Mons is a volcano on Mars
that was discovered by the Mariner 9
spacecraft during a global dust storm.
2. It is arguably the largest volcano in
the entire Solar System (about 100
times the volume of Mauna Loa).
3. This image was brought to us by
Hadamard matrices!
36 / 55
37. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Coding Theory
Quantum Computing
Much, Much More
Nirgal Vallis (1972)
(Credit: NASA)
1. Nirgal Vallis is a long and large river
valley network on Mars that was
discovered by the Mariner 9
spacecraft.
10pt
2. It served as one of the earliest pieces
of evidence for water on Mars.
3. This image was brought to us by
Hadamard matrices!
37 / 55
38. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Coding Theory
Quantum Computing
Much, Much More
Hadamard Code, Part 1
(Credit: Tilman Piesk)
1. This is the 64 × 32 matrix that Mariner 9 used to encode the
pictures it took of Mars. It is an example of a Hadamard code
which is a particular type of error-correcting code.
2. How were pictures encoded?
Each photoreceptor in the camera would measure the brightness
of a section of Mars and then output a grayscale value between 0
and 63 (e.g., 0=black, 63=white, etc.) Notice that the matrix on
the left has 64 rows ... each one is a codeword which represents
one particular grayscale value.
3. Why was an error-correcting code needed for this mission?
Because the low signal-to-noise ratio meant that a large fraction
of the transmitted bits of information would be corrupted.
Assuming a bit-failure probability of 10%, sending Mariner 9
without any error-correcting code would mean that about 47% of
each image would be in error.
38 / 55
39. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Coding Theory
Quantum Computing
Much, Much More
Hadamard Code, Part 2
(Credit: Tilman Piesk)
1. Let A be the matrix on the left and H =
1 1
1 −1
. Then
A =
−H32
H32
where
H32 = H⊗5 = H ⊗ H ⊗ H ⊗ H ⊗ H.
2. In particular, it is a (linear) [32, 6, 16]2-code.
2 = # of symbols in the alphabet (i.e., this is a binary code)
32 = length of each codeword (i.e., # of bits)
6 = dimension of the code
(so the total # of codewords is 26 = 64 which equals the
number of rows in A)
16 = minimum distance of the code (i.e., minimum # of
positions in which two distinct rows differ)
3. Since the minimum distance is 16, up to 7 bits of the original
message could be corrupted and yet the received word would
still be closer to the correct codeword than any other
codeword. Therefore, by minimum distance decoding, the
message could still be correctly decoded. This code is said to
be a 7-error correcting code.
39 / 55
40. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Coding Theory
Quantum Computing
Much, Much More
Hadamard Code, Part 3
Recall that if C is an (n, k, d)q code, then n is the length of each codeword, k is the
dimension of C, d is the minimum distance of C, and q is the number of symbols in the
alphabet.
Definition
The rate of an (n, k, d)q code is R =
k
n
.
For example, Mariner 9 used a (32, 6, 16)2 code that has a rate of R = 6
32 .
This means that a 6-bit message (grayscale value) is represented by a 32-bit codeword.
(All other things being equal, it is better to have a higher rate.)
Definition
An (n, k, d)q code is optimal if its rate, R, is as large as possible for given n, d, and q.
40 / 55
41. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Coding Theory
Quantum Computing
Much, Much More
Hadamard Code, Part 4
Theorem [Bose Shrikhande (1959) and Levenshtein (1964)]
The existence of a Hadamard matrix of order 4t implies the existence of the following
optimal codes:
1. (4t, log 8t, 2t)2
2. (4t − 1, log 4t, 2t)2
3. (4t − 1, log 8t, 2t − 1)2
4. (4t − 2, log 2t, 2t)2.
(Therefore, if the Hadamard conjecture is true, then the optimal codes above exist for
all positive integers, t.)
By Sylvester, there exists a Hadamard matrix of order 4 · 8 = 32. Here, we have t = 8.
It follows from (1) above that there must exist an optimal (32, 6, 16)2 code.
In fact, this is precisely the Hadamard code that was used by the Mariner 9 spacecraft!
41 / 55
42. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Coding Theory
Quantum Computing
Much, Much More
Quantum Computing, Part 1
Definition
A quantum computer is a machine that exploits quantum phenomena to store
information and perform computations.
Examples of quantum phenomena: the behavior of atoms, electrons, and photons (e.g.,
interference, entanglement, and superposition.)
(Credit: IBM Zurich Lab. See https://creativecommons.org/licenses/by-nd/2.0/legalcode.)
42 / 55
43. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Coding Theory
Quantum Computing
Much, Much More
Quantum Computing, Part 2
Definition
A quantum bit (or qubit) is a basic unit of information stored by a quantum computer.
When measured, a qubit is always found to be in precisely one of two possible basis
states. For concreteness, let us call these two basis states |0i and |1i. E.g., an electron’s
spin (down or up) or a photon’s polarization (left or right).
Key advantage of a quantum computer over a classical computer:
When a qubit is not being measured, it could be in any superposition (i.e., linear
combination) of the basis states:
|Ψi = α|0i + β|1i.
where α and β are each complex numbers whose square modulus equals the probability
of finding a qubit to be in the |0i or |1i state, respectively.
43 / 55
44. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Coding Theory
Quantum Computing
Much, Much More
Hadamard Gate
Definition
The Hadamard gate acts on a single qubit and is represented by the scaled-down
Hadamard matrix H = 1
√
2
1 1
1 −1
.
We can see what the Hadamard gate does to the basis states |0i =
1
0
and |1i =
0
1
by looking at the columns of H. To be sure:
H|0i =
1
√
2
1 1
1 −1
1
0
=
1
√
2
1
1
=
1
√
2
(|0i + |1i) (uniform superposition)
H|1i =
1
√
2
1 1
1 −1
0
1
=
1
√
2
1
−1
=
1
√
2
(|0i − |1i) (uniform superposition)
Essentially, the Hadamard gate turns a qubit into a state that can be determined by a
coin toss!
44 / 55
45. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Coding Theory
Quantum Computing
Much, Much More
Hadamard Transform
Definition
The Hadamard transform acts on n qubits and is represented by the scaled-down
Hadamard matrix H⊗n
.
2 qubits: H⊗2
|00i = (H ⊗ H)(|0i ⊗ |0i)
= H|0i ⊗ H|0i (Mixed-Product Prop.)
=
1
√
2
(|0i + |1i) ⊗
1
√
2
(|0i + |1i)
=
1
2
(|00i + |01i + |10i + |11i) (Distributive Prop.)
Observe that we have a uniform superposition of the 22
basis states.
45 / 55
46. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Coding Theory
Quantum Computing
Much, Much More
Hadamard Transform
Definition
The Hadamard transform acts on n qubits and is represented by the scaled-down
Hadamard matrix H⊗n
.
n qubits: H⊗n
|0 · · · 0i = (H ⊗ · · · ⊗ H)(|0i ⊗ · · · ⊗ |0i)
= H|0i ⊗ · · · ⊗ H|0i
=
1
√
2
(|0i + |1i) ⊗ · · · ⊗
1
√
2
(|0i + |1i)
=
1
√
2n
(|0 · · · 0i + |0 · · · 1i + · · · |1 · · · 1i)
Observe that we have a uniform superposition of the 2n
basis states.
Combined with interference and/or entanglement, this allows us to compute the value of
a function for 2n
different inputs in parallel (i.e., at the same time using only 1 circuit!)
46 / 55
48. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Coding Theory
Quantum Computing
Much, Much More
Grover’s Algorithm (Grover, 1996)
(Source: Jaden Pieper and Manuel E. Lladser)
1. This is a quantum database-searching algorithm (i.e., ”finding a needle in a
haystack”).
2. The circuit is initialized by putting n qubits in uniform superposition.
3. The algorithm exploits quantum superposition to evaluate a function at all 2n
different inputs in parallel.
4. The complexity is O(
√
n) versus O(n2
) for a classical algorithm.
48 / 55
49. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Coding Theory
Quantum Computing
Much, Much More
Deutsch’s Algorithm (Deutsch Jozsa 1992)
(Source: Jaden Pieper and Manuel E. Lladser)
1. This is an oracle demystifying algorithm (i.e., it determines whether a function is
constant [f (0) = f (1)] or balanced [f (0) 6= f (1)]).
2. Hadamard gates are used both before and after the main function (to exploit
quantum superposition and interference, respectively).
3. This algorithm is able to determine whether the function is constant or balanced
by evaluating the function only once!
4. This algorithm was the inspiration for Simon’s algorithm which, in turn, was the
inspiration for Shor’s prime-factorization algorithm.
49 / 55
50. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Coding Theory
Quantum Computing
Much, Much More
Much, Much More
Besides coding theory and quantum computing, Hadamard matrices have been used in
many fields such as:
1. Signal processing and data compression (e.g., CDMA, JPEG, and MPEG)
2. Design and analysis of experiments (statistics)
3. Nuclear magnetic resonance (NMR)
4. Mass spectrometry and crystallography
5. Graph theory and combinatorics
6. Frame theory (e.g., Tremain equiangular tight frames)
7. Much, much more
50 / 55
51. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Coding Theory
Quantum Computing
Much, Much More
Summary
1. Sylvester discovered Hadamard matrices in 1867.
2. Some of the earliest work was done by Hadamard and Scarpis in the 1890’s.
3. The Hadamard conjecture was first published in 1933 by Paley. It states that
Hadamard matrices exist for every order that is a multiple of 4.
4. The proof of the Hadamard conjecture would answer open questions in other fields
such as coding theory, graph theory, and design theory.
5. Using Sylvester’s constuction, Paley’s construction, and Williamson’s method, one
can prove Hadamard’s conjecture for all orders ≤ 100.
6. The smallest order (that is a multiple of 4) for which no Hadamard matrix is
known is currently 668. We have been stuck at this number since 2005!
7. There are many applications of Hadamard matrices in various fields including
coding theory, quantum computing, and statistics.
51 / 55
52. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Coding Theory
Quantum Computing
Much, Much More
References I
[1] J. J. Sylvester, Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tesselated pavements in two or
more colours, with applications to Newton‘s rule, ornamental tile-work, and the theory of numbers, Phil. Mag., Vol. 34
(1867), 461-475.
[2] J. Hadamard, Resolution d’une question relative aux determinants, Bull. Sci. Math., Vol. 17 (1893), 240-246.
[3] R.E.A.C. Paley, On orthogonal matrices, Journal of Mathematics and Physics 12 (1933), 311-320.
[4] R.C. Bose S.S. Shrikhande, A note on a result in the theory of code construction, Information and Control 2 (1959),
183-194.
[5] L.D. Baumert, S.W. Golomb, and M. Hall, Discovery of an Hadamard matrix of order 92, Bull. Amer. Math. Soc. 68:
(1962),237-238.
[6] V.I. Levenshtein, Application of the Hadamard matrices to a problem in coding, Problems of Cybernetics 5 (1964), 166-184.
[7] R.J. Turyn, An infinite class of Williamson matrices, J. Combin. Theory Ser. A 12 (1972), 319-321.
52 / 55
53. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Coding Theory
Quantum Computing
Much, Much More
References II
[8] A. Hedayat W. Wallis, Hadamard matrices and their applications, The Annals of Statistics 6(6) (1978), 1184-1238.
[9] H. Evangelaras, C. Koukouvinos, J. Seberry, Applications of Hadamard matrices, Journal of Telecommunications and
Information Technology, 2 (2003), 3-10.
[10] E.W. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC, New York (2003).
[11] J. Seberry, B.J. Wysocki, T.A. Wysocki, On some applications of Hadamard matrices, Metrika 62 (2005), 221-239.
[12] I.M. Wanless, Permanents of matrices of signed ones, Lin. and Multilin. Alg. 53(6) (2005), 427-433
[13] W.H. Holzmann, H. Kharaghani, B. Tayfeh-Rezaie, Williamson matrices up to order 59, Des. Codes Cryptogr. 46 (2008),
343-352
[14] D. McMahon, Quantum Computing Explained, Wiley, Hoboken (2008).
[15] T. Gowers, J. Barrow-Green, I. Leader, The Princeton Companion to Mathematics, Princeton University, Princeton (2008).
[16] R.J. Lipton K.W. Regan, Quantum Algorithms via Linear Algebra, MIT, Cambridge (2014).
53 / 55
54. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Coding Theory
Quantum Computing
Much, Much More
References III
[17] S. Kolpas, Mathematical treasure: a letter of James Joseph Sylvester to Leopold Kronecker, Convergence Oct. (2015).
[18] M. Fickus, J. Jasper, D. Mixon, J. Petersen Mixon, Tremain equiangular tight frames, Journal of Combinatorial Theory
Series A 153(C) (2018), 54-66.
[19] http://mathshistory.st-andrews.ac.uk/
[20] https://solarsystem.nasa.gov/missions/mariner-09/in-depth/
[21] https://mars.nasa.gov/gallery/atlas/olympus-mons.html
[22] https://www.britannica.com/place/Nirgal-Vallis
54 / 55
55. Hadamard
Matrices:
Truth
Consequences
Raymond Nguyen
Advisor: Peter
Casazza
The University of
Missouri
Math 8190
(Master’s Project)
Basic Theory of
Hadamard
Matrices
Hadamard Matrix
Constructions
Applications of
Hadamard
Matrices
Coding Theory
Quantum Computing
Much, Much More
Is This a Hadamard Matrix?
55 / 55
