### Ashley

```HADAMARD MATRICES AND THE
BY: ASHLEY REYNOLDS
• LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY
• SPECIAL MATRICES
DEFINITIONS
• MATRIX: IN MATHEMATICS, A MATRIX (PLURAL MATRICES) IS A RECTANGULAR ARRAY OF
NUMBERS, SYMBOLS, OR EXPRESSIONS, ARRANGED IN ROWS AND COLUMNS. THE INDIVIDUAL
ITEMS IN A MATRIX ARE CALLED ITS ELEMENTS OR ENTRIES.
DEFINITIONS
• SQUARE MATRIX: A MATRIX WITH THE SAME NUMBER OF ROWS AND COLUMNS.
• IDENTITY MATRIX: A SQUARE MATRIX IN WHICH ALL THE ELEMENTS OF THE PRINCIPAL
DIAGONAL ARE ONES AND ALL OTHER ELEMENTS ARE ZEROS.
DEFINITIONS
• TRANSPOSE OF A MATRIX:
• ORTHOGONAL:
HAVING THE SUM OF PRODUCTS OF CORRESPONDING ELEMENTS IN ANY TWO ROWS OR ANY TWO
COLUMNS EQUAL TO ONE IF THE ROWS OR COLUMNS ARE THE SAME AND EQUAL TO ZERO OTHERWISE
: HAVING A TRANSPOSE WITH WHICH THE PRODUCT EQUALS THE IDENTITY MATRIX.
DEFINITIONS
• DETERMINANT: THE DETERMINANT IS A VALUE ASSOCIATED WITH A SQUARE MATRIX. IT CAN
BE COMPUTED FROM THE ENTRIES OF THE MATRIX BY A SPECIFIC ARITHMETIC EXPRESSION,
WHILE OTHER WAYS TO DETERMINE ITS VALUE EXIST AS WELL. THE DETERMINANT PROVIDES
IMPORTANT INFORMATION ABOUT A MATRIX OF COEFFICIENTS OF A SYSTEM OF LINEAR
EQUATIONS, OR ABOUT A MATRIX THAT CORRESPONDS TO A LINEAR TRANSFORMATION OF A
VECTOR SPACE.
• A HADAMARD MATRIX, NAMED AFTER THE FRENCH
MATHEMATICIAN JACQUES HADAMARD, IS A SQUARE MATRIX
WHOSE ENTRIES ARE EITHER +1 OR −1 AND WHOSE ROWS ARE
MUTUALLY ORTHOGONAL. IN GEOMETRIC TERMS, THIS MEANS
THAT EVERY TWO DIFFERENT ROWS IN A HADAMARD MATRIX
REPRESENT TWO PERPENDICULAR VECTORS, WHILE IN
COMBINATORIAL TERMS, IT MEANS THAT EVERY TWO DIFFERENT
ROWS HAVE MATCHING ENTRIES IN EXACTLY HALF OF THEIR
COLUMNS AND MISMATCHED ENTRIES IN THE REMAINING
COLUMNS. IT IS A CONSEQUENCE OF THIS DEFINITION THAT THE
CORRESPONDING PROPERTIES HOLD FOR COLUMNS AS WELL AS
ROWS.
PROPERTIES OF A HARDAMARD MATRIX
• THE MOST IMPORTANT OPEN QUESTION IN THE THEORY OF HADAMARD MATRICES IS THAT OF
4K EXISTS FOR EVERY POSITIVE INTEGER K.
• A GENERALIZATION OF SYLVESTER’S CONSTRUCTION PROVES THAT IF HN AND HM ARE
HADAMARD MATRICES OF ORDER N AND M RESPECTIVELY, THEN
IS A HADAMARD MATRIX OF ORDER NM. THIS RESULT IS USED TO PRODUCE HADAMARD
MATRICES OF HIGHER ORDER ONCE THOSE OF SMALLER ORDERS ARE KNOWN.
• SYLVESTER'S 1867 CONSTRUCTION YIELDS HADAMARD MATRICES OF ORDER 1, 2, 4, 8, 16, 32, ETC. HADAMARD
MATRICES OF ORDERS 12 AND 20 WERE SUBSEQUENTLY CONSTRUCTED BY HADAMARD (IN 1893).
• IN 1933, RAYMOND PALEY DISCOVERED A CONSTRUCTION THAT PRODUCES A HADAMARD MATRIX OF ORDER
Q+1 WHEN Q IS ANY PRIME POWER THAT IS CONGRUENT TO 3 MODULO 4 AND THAT PRODUCES A HADAMARD
MATRIX OF ORDER 2(Q+1) WHEN Q IS A PRIME POWER THAT IS CONGRUENT TO 1 MODULO 4. HIS METHOD
USES FINITE FIELDS. THE HADAMARD CONJECTURE SHOULD PROBABLY BE ATTRIBUTED TO PALEY.
• THE SMALLEST ORDER THAT CANNOT BE CONSTRUCTED BY A COMBINATION OF SYLVESTER'S AND PALEY'S
METHODS IS 92. A HADAMARD MATRIX OF THIS ORDER WAS FOUND USING A COMPUTER BY BAUMERT,
GOLOMB, AND HALL IN 1962 AT JPL. THEY USED A CONSTRUCTION, DUE TO WILLIAMSON, THAT HAS YIELDED
MANY ADDITIONAL ORDERS. MANY OTHER METHODS FOR CONSTRUCTING HADAMARD MATRICES ARE NOW
KNOWN.