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 Matrix 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 Hadamard Matrix 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 square matrix 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 ”determinant”, 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 transpose 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 number 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 Raymond Paley (1933): “It seems probable that, whenever m is divisible by 4, it Applications of Hadamard is possible to construct an orthogonal matrix 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