Hadamard Matrices: Truth & Hadamard Matrices: Consequences Raymond Nguyen Advisor: Peter Truth & Consequences Casazza The University of Missouri Math 8190 (Master’s Project)

Basic Theory of Raymond Nguyen Hadamard Advisor: Peter Casazza Matrices Hadamard The University of Missouri Constructions Math 8190 (Master’s Project) Applications of Hadamard Matrices

Spring 2020

1 / 55 Hadamard Matrices: Outline Truth & Consequences

Raymond Nguyen Advisor: Peter Casazza The University of Missouri 1 Math 8190 Basic Theory of Hadamard Matrices (Master’s Project)

Basic Theory of Hadamard 2 Constructions Matrices Hadamard Matrix Constructions

Applications of 3 Hadamard Applications of Hadamard Matrices Matrices

2 / 55 Hadamard Matrices: Table of Contents Truth & Consequences

Raymond Nguyen Advisor: Peter 1 Basic Theory of Hadamard Matrices Casazza The University of Missouri Math 8190 Definition & Examples (Master’s Project)

Basic Theory of Properties Hadamard Matrices Definition & Examples The Hadamard Conjecture Properties The Hadamard Conjecture

Hadamard Matrix Constructions

2 Applications of Hadamard Matrix Constructions Hadamard Matrices

3 Applications of Hadamard Matrices

3 / 55 Hadamard Matrices: What is a Hadamard Matrix? Truth & Consequences

Raymond Nguyen Advisor: Peter Definition (Hadamard Matrix) Casazza The University of Missouri A H of order n whose entries are +1 or −1 is called a Math 8190 Hadamard matrix of order n provided its rows are pairwise orthogonal – (Master’s Project) i.e., Basic Theory of T Hadamard HH = n · In. (1) Matrices Definition & Examples Properties The Hadamard Conjecture Note that (1) implies that H has an inverse 1 HT . Consequently, its Hadamard Matrix n Constructions

columns are also pairwise orthogonal – i.e., Applications of Hadamard Matrices T H H = n · In. (2)

4 / 55 Hadamard Matrices: Examples Truth & Consequences

Raymond Nguyen Hadamard Matrices of Order 1, 2, 4, and 8: Advisor: Peter Casazza The University of   Missouri + + + + + + + + Math 8190   (Master’s Project)   + − + − + − + − + + + +   Basic Theory of   Hadamard + + − − + + − − Matrices           Definition & Examples + + + − + − Properties h i   + − − + + − − + + ,   ,   , and   The Hadamard Conjecture     + − + + − − + + + + − − − − Hadamard Matrix     Constructions         + − + − − + − + Applications of + − − +   Hadamard   Matrices + + − − − − + +   + − − + − + + −

5 / 55 Hadamard Matrices: “Anallagmatic Pavement” – J.J. Sylvester (1867) Truth & Consequences

Raymond Nguyen Advisor: Peter Casazza Hadamard Matrices of Order 1, 2, 4, and 8: 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

, , , and

6 / 55 Hadamard Matrices: James Joseph Sylvester (1814 – 1897) Truth & Consequences

Raymond Nguyen Advisor: Peter Casazza 1. Discovered Hadamard matrices The University of Missouri in 1867. Math 8190 (Master’s Project)

Basic Theory of 2. Coined the terms ”matrix”, Hadamard Matrices ””, and Definition & Examples Properties ”discriminant”. The Hadamard Conjecture

Hadamard Matrix Constructions 3. Served as the very first math Applications of professor at Johns Hopkins Hadamard Matrices

(Source: http://mathshistory.st-andrews.ac.uk/) University (1876-1883).

7 / 55 Hadamard Matrices: The Key Property of a Hadamard Matrix Truth & Consequences

Recall that the defining property of a Hadamard matrix is that Raymond Nguyen T Advisor: Peter its rows are orthogonal – i.e., HH = n · In. This property does not change if Casazza The University of we: Missouri Math 8190 (Master’s Project) 1. Take the of H. Basic Theory of Hadamard 2. Permute the rows of H. Matrices Definition & Examples Properties 3. Permute the columns of H. The Hadamard Conjecture

Hadamard Matrix 4. Multiply any row by -1. Constructions Applications of 5. Multiply any column by -1. Hadamard Matrices Definition (Hadamard Equivalence) Two Hadamard matrices are called equivalent if they differ by some sequence of the 5 operations listed above. 8 / 55 Hadamard Matrices: The Normal Form of a Hadamard Matrix Truth & Consequences

Raymond Nguyen Definition Advisor: Peter Casazza The University of A Hadamard matrix is called normalized if its first row and first column Missouri Math 8190 consist entirely of +1’s. (Master’s Project)

Basic Theory of Hadamard For example, here is a normalized Hadamard matrix of order 4: Matrices Definition & Examples   Properties + + + + The Hadamard Conjecture   Hadamard Matrix   Constructions + + − −   Applications of   Hadamard + − + − Matrices   + − − +

9 / 55 Hadamard Matrices: Normalization Truth & Consequences

Raymond Nguyen Advisor: Peter Fact Casazza The University of Every Hadamard matrix is equivalent to a normalized Hadamard matrix. Missouri Math 8190 (Master’s Project)

Given a Hadamard matrix, we can write it in normal form by negating every row Basic Theory of Hadamard and every column whose first element is −1. Matrices Definition & Examples Properties + − + + + − + + + + + + The Hadamard Conjecture       Hadamard Matrix + − − − + − − − + + − − Constructions   negate 3rd row   negate 2nd col     −−−−−−−−−→   −−−−−−−−→   Applications of − − − + + + + − + − + − Hadamard       Matrices + + − + + + − + + − − +

10 / 55 Hadamard Matrices: Properties of Normalized Hadamard Matrices Truth & Consequences

Raymond Nguyen Advisor: Peter Lemma Casazza The University of Suppose H is a normalized Hadamard matrix. With the exception of the first row Missouri (column), every row (column) of H has the following properties: Math 8190 (Master’s Project) 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). Basic Theory of Hadamard Matrices Definition & Examples Properties The Hadamard Conjecture + + + + + + + + + + + + + + + + Hadamard Matrix Constructions +  (1) + + − − (1) & (2) + + − − (1) & (2) + + − −   −−→   −−−−−→   −−−−−→   Applications of + + + − + − + − + − Hadamard         Matrices + + + + − − +

Up to equivalence, the matrix on the right is the unique Hadamard matrix of order 4.

11 / 55 Hadamard Matrices: How Many Distinct Hadamard Matrices of a Given Truth & Consequences

Order Exist? Raymond Nguyen Advisor: Peter Casazza The University of Missouri Math 8190 (Master’s Project) Up to equivalence, there is a unique Hadamard matrix of order m Basic Theory of for m = 1, 2, 4, 8, and 12. For m > 12, the of distinct Hadamard Hadamard Matrices matrices can be very large. Definition & Examples Properties The Hadamard Conjecture

order 1 2 4 8 12 16 20 24 28 32 Hadamard Matrix # 1 1 1 1 1 4 3 36 294 > 1 million Constructions Applications of Hadamard (Source: I.M. Wanless, 2005) Matrices

12 / 55 Hadamard Matrices: A Necessary Condition for Existence Truth & Consequences

Lemma Raymond Nguyen Advisor: Peter Suppose H is a normalized Hadamard matrix. With the exception of the first row Casazza The University of (column), every row (column) of H has the following properties: Missouri 1) Exactly half of the elements are +1’s and exactly half are −1’s. Math 8190 (Master’s Project) 2) Exactly half of the +1’s overlap with a +1 in any other given row (column). Basic Theory of Hadamard Matrices Theorem Definition & Examples Properties If a Hadamard matrix of order m exists, then m = 1, 2, or a multiple of 4. The Hadamard Conjecture

Hadamard Matrix Constructions   Applications of + + + + + ··· + + + + + + Hadamard + + ······ + + − − · · · · · · − − Matrices     + + + ··· + − · · · −  [+] , , or   + −  .   .  +

13 / 55 Hadamard Matrices: Question: Does there exist a Hadamard matrix for Truth & Consequences every order that is a multiple of 4? Raymond Nguyen Advisor: Peter Casazza The University of Missouri Math 8190 Answer: Nobody knows ... but go try this at home! (Master’s Project) Basic Theory of Hadamard The Hadamard Conjecture Matrices Definition & Examples A Hadamard matrix of order 4n exists for all positive integers n. Properties The Hadamard Conjecture

Hadamard Matrix Constructions (1933): “It seems probable that, whenever m is divisible by 4, it Applications of Hadamard is possible to construct an of order m composed of ±1, but Matrices the general theorem has every appearance of difficulty.”

14 / 55 Hadamard Matrices: Jacques Hadamard (1865 – 1963) Truth & Consequences

Raymond Nguyen Advisor: Peter Casazza The University of 1. Showed in 1893 that if M is a square Missouri matrix of order n whose entries are ±1, Math 8190 √ (Master’s Project) then det(M) ≤ ( n)n and that this bound is achieved iff M is a Hadamard matrix. Basic Theory of Hadamard (This is a special case of Hadamard’s Matrices maximal determinant problem.) Definition & Examples Properties 2. Proved that if a Hadamard matrix exists The Hadamard Conjecture Hadamard Matrix (i.e., if the bound above is achieved), then Constructions

its order must be 1, 2, or a multiple of 4. Applications of Hadamard 3. Was the first to construct Hadamard Matrices

(Source: http://mathshistory.st-andrews.ac.uk/) matrices of orders 12 and 20.

15 / 55 Hadamard Matrices: Table of Contents Truth & Consequences

Raymond Nguyen Advisor: Peter 1 Basic Theory of Hadamard Matrices Casazza The University of Missouri Math 8190 (Master’s Project) 2 Hadamard Matrix Constructions Basic Theory of Hadamard Sylvester’s Construction Matrices Hadamard Matrix Constructions Sylvester’s Construction Paley’s Construction Paley’s Construction Williamson’s Method

Applications of Williamson’s Method Hadamard Matrices

3 Applications of Hadamard Matrices

16 / 55 Hadamard Matrices: Truth & Consequences Definition Raymond Nguyen Advisor: Peter If A = (a ) is an m × p matrix and B = (b ) is a n × q matrix, then the Kronecker Casazza ij ij The University of product (or tensor product) A ⊗ B is the mn × pq matrix given by Missouri Math 8190   (Master’s Project) a11B a12B ··· a1pB a21B a22B ··· a2pB  Basic Theory of A ⊗ B =   Hadamard  .  Matrices  .  Hadamard Matrix am1B am2B ··· ampB Constructions Sylvester’s Construction Paley’s Construction Williamson’s Method

Properties Applications of T T T Hadamard 1) Transpose Property:(A ⊗ B) = A ⊗ B . Matrices

2) Mixed-Product Property:(A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD) (assuming A, B, C, and D are of the correct size for .)

17 / 55 Hadamard Matrices: Sylvester’s Construction, Part 1 Truth & Consequences

Raymond Nguyen Advisor: Peter Theorem (Sylvester) Casazza The University of If H1 is a Hadamard matrix of order m and H2 is a Hadamard matrix of Missouri Math 8190 order n then H1 ⊗ H2 is a Hadamard matrix of order mn. (Master’s Project)

Basic Theory of Hadamard Proof: Matrices Hadamard Matrix Constructions T T T (H1 ⊗ H2)(H1 ⊗ H2) = (H1 ⊗ H2)(H ⊗ H ) (Transpose Prop.) Sylvester’s Construction 1 2 Paley’s Construction T T Williamson’s Method = (H1H ) ⊗ (H2H ) (Mixed-Product Prop.) 1 2 Applications of Hadamard = (m · Im) ⊗ (n · In) (Hadamard Matrices) Matrices

= mn · Imn.

18 / 55 Hadamard Matrices: Sylvester’s Construction, Part 2 Truth & Consequences

Raymond Nguyen Advisor: Peter Corollary (Sylvester) Casazza The University of n Missouri There exists a Hadamard matrix of order 2 for every positive integer n. Math 8190 (Master’s Project)

Basic Theory of Proof: We already know that there exists a Hadamard matrix of order 2: Hadamard Matrices   Hadamard Matrix + + Constructions H =   . Sylvester’s Construction Paley’s Construction + − Williamson’s Method Applications of Hadamard Matrices Now repeatedly take tensor powers of H to obtain a Hadamard matrix of order 2n for any positive integer n.

19 / 55 Hadamard Matrices: Examples Using Sylvester’s Construction Truth & Consequences

Raymond Nguyen Hadamard matrices of order 2, 4, 8, ... : 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 H + − − +   + + − − − − + + Applications of =   Hadamard H⊗H + − − + − + + − Matrices = H⊗H⊗H

20 / 55 Hadamard Matrices: What We Know So Far Truth & Consequences

Raymond Nguyen Advisor: Peter Casazza The University of Missouri The Hadamard Conjecture Math 8190 (Master’s Project) A Hadamard matrix of order 4n exists for all positive integers n. Basic Theory of Hadamard Matrices

Multiples of 4 that are less than or equal to 100: Hadamard Matrix Constructions 4,8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, Sylvester’s Construction Paley’s Construction 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100 Williamson’s Method Applications of k Hadamard Sylvester’s construction[ m = 2 or m = m1m2 where m1 and m2 are Hadamard orders] Matrices

21 / 55 Hadamard Matrices: Prime Time Truth & Consequences

Raymond Nguyen Advisor: Peter Theorem (Umberto Scarpis, 1898) Casazza The University of Suppose p is a . Then we have the following: Missouri Math 8190 1) If p ≡ 3 (mod 4), then there is a Hadamard matrix of order p + 1. (Master’s Project)

2) If p ≡ 1 (mod 4), then there is a Hadamard matrix of order 2(p + 1). Basic Theory of Hadamard Matrices

Hadamard Matrix In 1933, Paley discovered two constructions which generalize the work of Scarpis. Constructions Sylvester’s Construction Paley’s Construction Theorem (Paley, 1933) Williamson’s Method Applications of Suppose p is a prime number and α > 0. Then we have the following: Hadamard 1) If pα ≡ 3 (mod 4), then there is a Hadamard matrix of order pα + 1. Matrices 2) If pα ≡ 1 (mod 4), then there is a Hadamard matrix of order 2(pα + 1).

22 / 55 Hadamard Matrices: Paley’s Construction, Type 1 Truth & Consequences

Paley’s 1st Theorem Raymond Nguyen Advisor: Peter Suppose p is a prime number and α > 0. Then we have the following: Casazza If pα ≡ 3 (mod 4), then there is a Hadamard matrix of order m = pα + 1. The University of Missouri Math 8190 (Master’s Project) α Sketch of Proof: Label the elements of GF(p ) as a0, a1, a2,... in some order. α Let Q = (qij ) be a matrix of order p whose entries are given by qij = χ(ai − aj ) where χ is the Basic Theory of quadratic character on GF(pα). That is, Hadamard Matrices  0, if b = 0 Hadamard Matrix  Constructions α χ(b) = +1, if b is a non-zero perfect square in GF(p ) . Sylvester’s Construction −1, if b is not a perfect square in GF(pα) Paley’s Construction Williamson’s Method h 0 1Ti Applications of Form the m × m matrix C = and write H = Im + C. Hadamard −1 Q Matrices Then H is a Hadamard matrix of order m = pα + 1. [Note: The key to the proof is that C is an anti-symmetric . That is, C = −C T (anti-symmetric). Also, C has all 0’s along the diagonal, ±1 elsewhere, and T CC = (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 Hadamard Matrices: Example Using Paley’s 1st Theorem Truth & Consequences

Let’s construct a Hadamard matrix of order 12 (= 11 + 1). Raymond Nguyen α α Advisor: Peter Here we have p = 11 ≡ 3 (mod 4) and p + 1 = 12 so such a matrix must exist. Casazza The University of Write the elements of GF (11) as a0 = 0, a1 = 1, a2 = 2,... . Missouri Math 8190 Observe that the perfect squares of GF (11) are 1, 3, 4, 5, and 9. (Master’s Project) That is, 12 ≡ 1, 52 ≡ 3, 22 ≡ 4, 42 ≡ 5, and 32 ≡ 9. Basic Theory of Then the quadratic character matrix Q and the Hadamard matrix H are: Hadamard Matrices  0 − + − − − + + + − +   ++++++++++++  − + − + − − − + + + − + Hadamard Matrix + 0 − + − − − + + + − Constructions − + 0 − + − − − + + +  − + + − + − − − + + + −    − − + + − + − − − + + + Sylvester’s Construction  + − + 0 − + − − − + +     − + − + + − + − − − + +  Paley’s Construction  + + − + 0 − + − − − +     − + + − + + − + − − − +  Williamson’s Method Q = + + + − + 0 − + − − − −→ H =  .    − + + + − + + − + − − −  Applications of  − + + + − + 0 − + − −   − − + + + − + + − + − −   − − + + + − + 0 − + −    Hadamard    − − − + + + − + + − + −  Matrices  − − − + + + − + 0 − +   − − − − + + + − + + − +  + − − − + + + − + 0 − − + − − − + + + − + + − − + − − − + + + − + 0 − − + − − − + + + − + + Observe that Q is a . That is, each row is obtained from the row above it by a cyclic permutation. 24 / 55 Hadamard Matrices: Paley’s Construction, Type 2 Truth & Consequences

Raymond Nguyen Paley’s 2nd Theorem Advisor: Peter Suppose p is a prime number and α > 0. Then we have the following: Casazza α α The University of If p ≡ 1 (mod 4), then there is a Hadamard matrix of order m = 2(p + 1). Missouri Math 8190 (Master’s Project) α Sketch of Proof: As in Paley’s first theorem, let Q = (qij ) be a matrix of order p whose entries are Basic Theory of given by qij = χ(ai − aj ). Hadamard Matrices 0 1T Let n = pα + 1. Form the n × n matrix C = Hadamard Matrix 1 Q Constructions Sylvester’s Construction + + + − Paley’s Construction and this time write H = C ⊗ + I ⊗ . Williamson’s Method + − n − − Applications of α Hadamard Then H is a Hadamard matrix of order m = 2n = 2(p + 1). Matrices [Note: The key to the proof is that C is a symmetric conference matrix. That is, C = C T (symmetric). Also, C has all 0’s along the diagonal, ±1 elsewhere, and T CC = (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 Hadamard Matrices: Example using Paley’s 2nd Theorem Truth & Consequences Let’s construct another Hadamard matrix of order 12 [= 2(5 + 1)] ... this time using Paley’s second Raymond Nguyen theorem. Advisor: Peter α α Casazza Here we have p = 5 ≡ 1 (mod 4) and 2(p + 1) = 12 so such a matrix must exist. The University of Missouri Write the elements of GF(5) as a0 = 0, a1 = 1, a2 = 2,... . Math 8190 Observe that the perfect squares of GF(5) are 1 and 4. (Master’s Project) 2 2 That is, 1 ≡ 1 and 2 ≡ 4. Basic Theory of Hadamard Then the quadratic character matrix Q, the conference matrix C, and the Hadamard matrix H are: Matrices  + − + + + + + + + + + +  Hadamard Matrix − − + − + − + − + − + − Constructions + + + − + + − − − − + + Sylvester’s Construction  0 + + + + +   + − − − + − − + − + + −  Paley’s Construction 0 + − − + " # + 0 + − − +  + + + + + − + + − − − −  + 0 + − − Williamson’s Method + + 0 + − −  + − + − − − + − − + − +  − + 0 + −     Q = −→ C = + − + 0 + − −→ H = + + − − + + + − + + − − . Applications of − − + 0 +     + − − + 0 + + − − + + − − − + − − + Hadamard + − − + 0   + + − − + 0  + + − − − − + + + − + +  Matrices  + − − + − + + − − − + −  + + + + − − − − + + + − + − + − − + − + + − − −

  + +   + −   Recall: H = C ⊗ + − + In ⊗ − − .

26 / 55 Hadamard Matrices: Raymond E.A.C. Paley (January 1907 - ) 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

(Source: http://mathshistory.st-andrews.ac.uk/)

[Credit: The Times (Canada)]

27 / 55 Hadamard Matrices: What We Know So Far Truth & Consequences

Raymond Nguyen Advisor: Peter The Hadamard Conjecture Casazza The University of A Hadamard matrix of order 4n exists for all positive integers n. Missouri Math 8190 (Master’s Project)

Multiples of 4 that are less than or equal to 100: Basic Theory of Hadamard Matrices

4,8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, Hadamard Matrix 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100 Constructions Sylvester’s Construction Paley’s Construction k Sylvester’s construction[ m = 2 or m = m1m2 where m1 and m2 are Hadamard orders] Williamson’s Method α α Applications of Paley’s construction, type 1[ m = p + 1 where p ≡ 3 (mod 4)] Hadamard Matrices 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 Hadamard Matrices: A Path Forward Truth & Theorem (John Williamson, 1944) Consequences Raymond Nguyen Advisor: Peter If A, B, C and D are square (1,-1) matrices of order n which satisfy Casazza T T T T 1. AA + BB + CC + DD = 4n · In and The University of 2. XY T = YX T Missouri Math 8190 for any two distinct matrices X, Y ∈ {A, B, C, D}, (Master’s Project)  ABCD  −BA −DC then H = is a Hadamard matrix of order 4n. Basic Theory of −CDA −B Hadamard −D −CBA Matrices Hadamard Matrix Constructions T Sylvester’s Construction Proof: Let’s consider M = HH as a with n × n blocks. Observe: Paley’s Construction Williamson’s Method

T T T T (1) Applications of 1. Mii = AA + BB + CC + DD = 4n · In. Hadamard Matrices

T T T T (2) 2. M1,2 = −AB + BA − CD + DC = 0 + 0 = 0. (similarly for other i 6= j)

 4n·In 0 0 0  Therefore, we have M = HHT = 0 4n·In 0 0 = 4n · I . 0 0 4n·In 0 4n 0 0 0 4n·In 29 / 55 Hadamard Matrices: Williamson’s Method Truth & Consequences Theorem (John Williamson, 1944) Raymond Nguyen Advisor: Peter If A, B, C, and D are square matrices of order n whose entries are ±1 which satisfy Casazza T T T T The University of 1. AA + BB + CC + DD = 4n · In and Missouri 2. XY T = YX T Math 8190 for any two distinct matrices X, Y ∈ {A, B, C, D}, (Master’s Project)  ABCD  Basic Theory of −BA −DC then H = is a Hadamard matrix of order 4n. Hadamard −CDA −B Matrices −D −CBA Hadamard Matrix Constructions Sylvester’s Construction Paley’s Construction To further narrow the search for Hadamard matrices, Williamson made 2 simplifying assumptions: Williamson’s Method

Applications of 1. A, B, C, and D are symmetric. (This reduces the second condition XY T = YX T to saying that Hadamard A, B, C, and D commute.) Matrices

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 Hadamard Matrices: Breakthroughs Using Williamson’s Method Truth & Consequences

Raymond Nguyen Advisor: Peter Using Williamson’s method, many previously unknown Hadamard matrices have Casazza been discovered for various orders. We call these Williamson matrices. The University of Missouri Math 8190 (Master’s Project)

1. Williamson (1944) used to find Williamson matrices of order Basic Theory of 148 (= 4 · 37) and 172 (= 4 · 43). Hadamard Matrices

Hadamard Matrix 2. Baumert, Golomb, and Hall (1962) used a computer to find a Williamson Constructions Sylvester’s Construction matrix of order 92 (= 4 · 23). (This is the missing order from three slides Paley’s Construction ago.) Williamson’s Method Applications of Hadamard 3. Turyn (1972) found an infinite class of Williamson matrices. That is, Matrices if n is odd and 2n − 1 is a prime power, then there is a Williamson matrix of order 4n.

31 / 55 Hadamard Matrices: Hmm ... That’s Odd! Truth & Consequences

Notice that a lot of effort has gone into finding Hadamard matrices of order 4n where n Raymond Nguyen is odd. Here’s why: Advisor: Peter Casazza The University of Missouri Theorem Math 8190 Suppose that for every odd n > 0, there exists a Hadamard matrix of order 4n. (Master’s Project)

Then there exists a Hadamard matrix of order m = 4n for every n > 0. (I.e., then the Basic Theory of Hadamard conjecture is true.) Hadamard Matrices

Hadamard Matrix i Constructions Proof: Let n > 0. Write m = 4n = 2 j for some i ≥ 2 and j odd. Sylvester’s Construction Paley’s Construction Williamson’s Method

1. If i = 2 then m = 4j where j is odd. By assumption, there exists a Hadamard Applications of Hadamard matrix of order m = 4j. Matrices 2. If i > 2 then m = 2k 4j for some k > 0. By Sylvester’s corollary, there exists a k Hadamard matrix H1 of order 2 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 Hadamard Matrices: Question: Do Williamson Matrices of Order 4n Exist Truth & Consequences for Every Odd Positive Integer n? Raymond Nguyen Advisor: Peter Answer Casazza The University of Missouri Math 8190 : If the answer is “yes”, then the Hadamard conjecture is true. (Master’s Project)

Unfortunately, the answer is “no”! Basic Theory of Hadamard Matrices

1. Dokovic (1993) showed that there are no Williamson matrices of order Hadamard Matrix Constructions 4n for n = 35. Sylvester’s Construction Paley’s Construction 2. Holzmann et. al (2007) showed that there are no Williamson matrices Williamson’s Method Applications of of order 47, 53, or 59. Hadamard Matrices 3. Therefore, it is not possible to prove the Hadamard conjecture solely by using Williamson matrices.

33 / 55 Hadamard Matrices: Table of Contents Truth & Consequences

Raymond Nguyen Advisor: Peter 1 Basic Theory of Hadamard Matrices Casazza The University of Missouri Math 8190 (Master’s Project) 2 Hadamard Matrix Constructions Basic Theory of Hadamard Matrices

Hadamard Matrix 3 Applications of Hadamard Matrices Constructions Applications of Hadamard Coding Theory Matrices Coding Theory Quantum Computing Much, Much More

Much, Much More

34 / 55 Hadamard Matrices: Mariner 9 (May 30, 1971 – October 27, 1972) Truth & Consequences

Raymond Nguyen Advisor: Peter Casazza The University of Missouri Math 8190 (Master’s Project)

Basic Theory of Hadamard Matrices (Credit: NASA) Hadamard Matrix Constructions

1. Mariner 9 became the first spacecraft to orbit another planet (Mars). Applications of Hadamard Matrices 2. It mapped over 85% of the Martian surface and sent back over 7,000 pictures Coding Theory Quantum Computing including images of Olympus Mons, Valles Marineris, and Phobos & Deimos. Much, Much More

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 Hadamard Matrices: Olympus Mons (1972) Truth & Consequences

Raymond Nguyen Advisor: Peter Casazza The University of Missouri 1. Olympus Mons is a volcano on Mars Math 8190 (Master’s Project) that was discovered by the Mariner 9 spacecraft during a global dust storm. Basic Theory of Hadamard Matrices 2. It is arguably the largest volcano in Hadamard Matrix the entire Solar System (about 100 Constructions

times the volume of Mauna Loa). Applications of Hadamard Matrices 3. This image was brought to us by Coding Theory Hadamard matrices! Quantum Computing Much, Much More (Credit: NASA)

36 / 55 Hadamard Matrices: Nirgal Vallis (1972) Truth & Consequences

Raymond Nguyen Advisor: Peter Casazza The University of Missouri 1. Nirgal Vallis is a long and large river Math 8190 valley network on Mars that was (Master’s Project) discovered by the Mariner 9 Basic Theory of spacecraft. Hadamard 10pt Matrices Hadamard Matrix 2. It served as one of the earliest pieces Constructions Applications of of evidence for water on Mars. Hadamard Matrices (Credit: NASA) 3. This image was brought to us by Coding Theory Quantum Computing Hadamard matrices! Much, Much More

37 / 55 Hadamard Matrices: , Part 1 Truth & Consequences

Raymond Nguyen Advisor: Peter 1. This is the 64 × 32 matrix that Mariner 9 used to encode the Casazza pictures it took of Mars. It is an example of a Hadamard code The University of which is a particular type of error-correcting code. Missouri Math 8190 (Master’s Project) 2. How were pictures encoded? Each photoreceptor in the camera would measure the brightness Basic Theory of Hadamard of a section of Mars and then output a grayscale value between 0 Matrices 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 Hadamard Matrix Constructions one particular grayscale value. Applications of Hadamard 3. Why was an error-correcting code needed for this mission? Matrices Because the low signal-to-noise ratio meant that a large fraction Coding Theory Quantum Computing of the transmitted bits of information would be corrupted. Much, Much More 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. (Credit: Tilman Piesk)

38 / 55 Hadamard Matrices: Hadamard Code, Part 2 Truth & Consequences

 1 1  Raymond Nguyen 1. Let A be the matrix on the left and H = 1 −1 . Then Advisor: Peter  −H32  A = H where Casazza 32 The University of ⊗5 H32 = H = H ⊗ H ⊗ H ⊗ H ⊗ H. Missouri Math 8190 (Master’s Project) 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) Basic Theory of 32 = length of each codeword (i.e., # of bits) Hadamard Matrices 6 = dimension of the code (so the total # of codewords is 26 = 64 which equals the Hadamard Matrix number of rows in A) Constructions 16 = minimum distance of the code (i.e., minimum # of Applications of positions in which two distinct rows differ) Hadamard Matrices Coding Theory 3. Since the minimum distance is 16, up to 7 bits of the original Quantum Computing message could be corrupted and yet the received word would Much, Much More 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 (Credit: Tilman Piesk) be a 7-error correcting code.

39 / 55 Hadamard Matrices: Hadamard Code, Part 3 Truth & Consequences

Raymond Nguyen Recall that if C is an (n, k, d)q code, then n is the length of each codeword, k is the Advisor: Peter dimension of C, d is the minimum distance of C, and q is the number of symbols in the Casazza The University of alphabet. Missouri Math 8190 Definition (Master’s Project) k Basic Theory of The rate of an (n, k, d) code is R = . Hadamard q n Matrices Hadamard Matrix Constructions 6 For example, Mariner 9 used a (32, 6, 16)2 code that has a rate of R = . Applications of 32 Hadamard Matrices This means that a 6-bit message (grayscale value) is represented by a 32-bit codeword. Coding Theory (All other things being equal, it is better to have a higher rate.) Quantum Computing Much, Much More 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 Hadamard Matrices: Hadamard Code, Part 4 Truth & Consequences

Theorem [Bose & Shrikhande (1959) and Levenshtein (1964)] Raymond Nguyen Advisor: Peter The existence of a Hadamard matrix of order 4t implies the existence of the following Casazza The University of optimal codes: Missouri Math 8190 (Master’s Project) 1. (4t, log 8t, 2t)2 Basic Theory of 2. (4t − 1, log 4t, 2t)2 Hadamard Matrices 3. (4t − 1, log 8t, 2t − 1)2 Hadamard Matrix 4. (4t − 2, log 2t, 2t)2. Constructions Applications of (Therefore, if the Hadamard conjecture is true, then the optimal codes above exist for Hadamard Matrices all positive integers, t.) Coding Theory Quantum Computing Much, Much More 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 Hadamard Matrices: Quantum Computing, Part 1 Truth & Consequences

Definition Raymond Nguyen Advisor: Peter A quantum computer is a machine that exploits quantum phenomena to store Casazza The University of information and perform computations. Missouri Math 8190 (Master’s Project)

Examples of quantum phenomena: the behavior of atoms, electrons, and photons (e.g., Basic Theory of interference, entanglement, and superposition.) Hadamard Matrices

Hadamard Matrix Constructions

Applications of Hadamard Matrices Coding Theory Quantum Computing Much, Much More

(Credit: IBM Zurich Lab. See https://creativecommons.org/licenses/by-nd/2.0/legalcode.)

42 / 55 Hadamard Matrices: Quantum Computing, Part 2 Truth & Consequences

Raymond Nguyen Definition Advisor: Peter Casazza A quantum bit (or qubit) is a basic unit of information stored by a quantum computer. The University of Missouri Math 8190 (Master’s Project) 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 Basic Theory of Hadamard spin (down or up) or a photon’s polarization (left or right). Matrices Hadamard Matrix Key advantage of a quantum computer over a classical computer: Constructions Applications of When a qubit is not being measured, it could be in any superposition (i.e., linear Hadamard Matrices combination) of the basis states: Coding Theory Quantum Computing |Ψi = α|0i + β|1i. Much, Much More

where α and β are each complex whose square modulus equals the probability of finding a qubit to be in the |0i or |1i state, respectively.

43 / 55 Hadamard Matrices: Hadamard Gate Truth & Consequences

Definition Raymond Nguyen Advisor: Peter The Hadamard gate acts on a single qubit and is represented by the scaled-down Casazza   The University of 1 1 Missouri Hadamard matrix H = √1 . 2 Math 8190 1 −1 (Master’s Project)

Basic Theory of 1 0 Hadamard We can see what the Hadamard gate does to the basis states |0i = and |1i = Matrices 0 1 Hadamard Matrix by looking at the columns of H. To be sure: Constructions

Applications of 1 1 1  1 1 1 1 Hadamard H|0i = √ = √ = √ (|0i + |1i) (uniform superposition) Matrices 2 1 −1 0 2 1 2 Coding Theory Quantum Computing 1 1 1  0 1  1  1 Much, Much More H|1i = √ = √ = √ (|0i − |1i) (uniform superposition) 2 1 −1 1 2 −1 2 Essentially, the Hadamard gate turns a qubit into a state that can be determined by a coin toss! 44 / 55 Hadamard Matrices: Truth & Consequences

Raymond Nguyen Definition Advisor: Peter Casazza The Hadamard transform acts on n qubits and is represented by the scaled-down The University of ⊗n Missouri Hadamard matrix H . Math 8190 (Master’s Project)

Basic Theory of Hadamard Matrices ⊗2 2 qubits: H |00i = (H ⊗ H)(|0i ⊗ |0i) Hadamard Matrix Constructions = H|0i ⊗ H|0i (Mixed-Product Prop.) Applications of 1 1 Hadamard = √ (|0i + |1i) ⊗ √ (|0i + |1i) Matrices 2 2 Coding Theory Quantum Computing 1 Much, Much More = (|00i + |01i + |10i + |11i) (Distributive Prop.) 2 Observe that we have a uniform superposition of the 22 basis states.

45 / 55 Hadamard Matrices: Hadamard Transform Truth & Consequences

Definition Raymond Nguyen Advisor: Peter The Hadamard transform acts on n qubits and is represented by the scaled-down Casazza Hadamard matrix H⊗n. The University of Missouri Math 8190 (Master’s Project)

Basic Theory of Hadamard ⊗n n qubits: H |0 ··· 0i = (H ⊗ · · · ⊗ H)(|0i ⊗ · · · ⊗ |0i) Matrices Hadamard Matrix = H|0i ⊗ · · · ⊗ H|0i Constructions

1 1 Applications of = √ (|0i + |1i) ⊗ · · · ⊗ √ (|0i + |1i) Hadamard 2 2 Matrices 1 Coding Theory = √ (|0 ··· 0i + |0 ··· 1i + · · · |1 ··· 1i) Quantum Computing 2n Much, Much More 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 Hadamard Matrices: Hadamard Transform Matrices Truth & Consequences

Raymond Nguyen Hadamard transform on 1 qubit, 2 qubits, 3 qubits, ... : Advisor: Peter Casazza The University of + + + + + + + + Missouri Math 8190 + − + − + − + − (Master’s Project)     + + + +   + + − − + + − − Basic Theory of     Hadamard       Matrices 1 + + 1 + − + − 1 + − − + + − − +     √  , √  , √  ,... Hadamard Matrix 2 + − 22 + + − − 23 + + + + − − − − Constructions         Applications of = + − + − − + − + Hadamard H + − − +   Matrices + + − − − − + + Coding Theory

=   Quantum Computing H⊗H + − − + − + + − Much, Much More = H⊗H⊗H

47 / 55 Hadamard Matrices: Grover’s Algorithm (Grover, 1996) Truth & Consequences

Raymond Nguyen Advisor: Peter Casazza The University of Missouri Math 8190 (Master’s Project)

(Source: Jaden Pieper and Manuel E. Lladser) Basic Theory of Hadamard Matrices 1. This is a quantum database-searching algorithm (i.e., ”finding a needle in a Hadamard Matrix Constructions haystack”). Applications of Hadamard 2. The circuit is initialized by putting n qubits in uniform superposition. Matrices Coding Theory n 3. The algorithm exploits quantum superposition to evaluate a function at all 2 Quantum Computing different inputs in parallel. Much, Much More √ 4. The complexity is O( n) versus O(n2) for a classical algorithm.

48 / 55 Hadamard Matrices: Deutsch’s Algorithm (Deutsch & Jozsa 1992) Truth & Consequences

Raymond Nguyen Advisor: Peter Casazza The University of Missouri Math 8190 (Master’s Project)

(Source: Jaden Pieper and Manuel E. Lladser) Basic Theory of Hadamard Matrices 1. This is an oracle demystifying algorithm (i.e., it determines whether a function is Hadamard Matrix Constructions constant [f (0) = f (1)] or balanced [f (0) 6= f (1)]). Applications of Hadamard 2. Hadamard gates are used both before and after the main function (to exploit Matrices quantum superposition and interference, respectively). Coding Theory Quantum Computing 3. This algorithm is able to determine whether the function is constant or balanced Much, Much More 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 Hadamard Matrices: Much, Much More Truth & Consequences

Raymond Nguyen Advisor: Peter Besides coding theory and quantum computing, Hadamard matrices have been used in Casazza The University of many fields such as: Missouri Math 8190 (Master’s Project) 1. Signal processing and data compression (e.g., CDMA, JPEG, and MPEG) Basic Theory of Hadamard 2. Design and analysis of experiments () Matrices

Hadamard Matrix 3. Nuclear magnetic resonance (NMR) Constructions 4. Mass spectrometry and crystallography Applications of Hadamard Matrices 5. and Coding Theory Quantum Computing 6. Frame theory (e.g., Tremain equiangular tight frames) Much, Much More 7. Much, much more

50 / 55 Hadamard Matrices: Summary Truth & Consequences

Raymond Nguyen Advisor: Peter 1. Sylvester discovered Hadamard matrices in 1867. Casazza The University of 2. Some of the earliest work was done by Hadamard and Scarpis in the 1890’s. Missouri Math 8190 3. The Hadamard conjecture was first published in 1933 by Paley. It states that (Master’s Project)

Hadamard matrices exist for every order that is a multiple of 4. Basic Theory of Hadamard 4. The proof of the Hadamard conjecture would answer open questions in other fields Matrices such as coding theory, graph theory, and design theory. Hadamard Matrix Constructions

5. Using Sylvester’s constuction, Paley’s construction, and Williamson’s method, one Applications of Hadamard can prove Hadamard’s conjecture for all orders ≤ 100. Matrices Coding Theory 6. The smallest order (that is a multiple of 4) for which no Hadamard matrix is Quantum Computing known is currently 668. We have been stuck at this number since 2005! Much, Much More 7. There are many applications of Hadamard matrices in various fields including coding theory, quantum computing, and statistics.

51 / 55 Hadamard Matrices: ReferencesI Truth & Consequences

Raymond Nguyen [1] J. J. Sylvester, Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tesselated pavements in two or Advisor: Peter more colours, with applications to Newton‘s rule, ornamental tile-work, and the theory of numbers, Phil. Mag., Vol. 34 Casazza (1867), 461-475. The University of Missouri Math 8190 (Master’s Project) [2] J. Hadamard, Resolution d’une question relative aux , Bull. Sci. Math., Vol. 17 (1893), 240-246.

Basic Theory of [3] R.E.A.C. Paley, On orthogonal matrices, Journal of and Physics 12 (1933), 311-320. Hadamard Matrices

Hadamard Matrix [4] R.C. Bose & S.S. Shrikhande, A note on a result in the theory of code construction, Information and Control 2 (1959), Constructions 183-194. Applications of Hadamard [5] L.D. Baumert, S.W. Golomb, and M. Hall, Discovery of an Hadamard matrix of order 92, Bull. Amer. Math. Soc. 68: Matrices (1962),237-238. Coding Theory Quantum Computing Much, Much More [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 Hadamard Matrices: ReferencesII Truth & [8] A. Hedayat & W. Wallis, Hadamard matrices and their applications, The Annals of Statistics 6(6) (1978), 1184-1238. Consequences Raymond Nguyen Advisor: Peter [9] H. Evangelaras, C. Koukouvinos, & J. Seberry, Applications of Hadamard matrices, Journal of Telecommunications and Casazza Information Technology, 2 (2003), 3-10. The University of Missouri Math 8190 [10] E.W. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC, New York (2003). (Master’s Project)

Basic Theory of [11] J. Seberry, B.J. Wysocki, & T.A. Wysocki, On some applications of Hadamard matrices, Metrika 62 (2005), 221-239. Hadamard Matrices

[12] I.M. Wanless, Permanents of matrices of signed ones, Lin. and Multilin. Alg. 53(6) (2005), 427-433 Hadamard Matrix Constructions

Applications of [13] W.H. Holzmann, H. Kharaghani, & B. Tayfeh-Rezaie, Williamson matrices up to order 59, Des. Codes Cryptogr. 46 (2008), Hadamard 343-352 Matrices Coding Theory Quantum Computing [14] D. McMahon, Quantum Computing Explained, Wiley, Hoboken (2008). Much, Much More

[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 Hadamard Matrices: ReferencesIII Truth & Consequences

Raymond Nguyen Advisor: Peter Casazza [17] S. Kolpas, Mathematical treasure: a letter of James Joseph Sylvester to Leopold Kronecker, Convergence Oct. (2015). The University of Missouri Math 8190 [18] M. Fickus, J. Jasper, D. Mixon, & J. Petersen Mixon, Tremain equiangular tight frames, Journal of Combinatorial Theory (Master’s Project) Series A 153(C) (2018), 54-66.

Basic Theory of Hadamard [19] http://mathshistory.st-andrews.ac.uk/ Matrices

Hadamard Matrix [20] https://solarsystem.nasa.gov/missions/mariner-09/in-depth/ Constructions Applications of Hadamard [21] https://mars.nasa.gov/gallery/atlas/olympus-mons.html Matrices Coding Theory Quantum Computing [22] https://www.britannica.com/place/Nirgal-Vallis Much, Much More

54 / 55 Hadamard Matrices: Is This a Hadamard Matrix? 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

55 / 55