Hadamard's maximal determinant problem
Hadamard's Maximal Determinant Problem is about finding the biggest number you can get when you multiply some numbers together. But it's not just any numbers - they have to be in a special kind of square pattern.
A square pattern is like a checkerboard, with the same number of rows and columns. Imagine a 3x3 checkerboard with 9 squares. Now, what if we put a number in each square so that no two numbers are the same and the absolute value of the difference between any two numbers is one or less? This means that if we subtract one of the numbers from another number, the answer will be no bigger than one.
For example, let's say we put a 2 in the top left square. The only numbers we can put in the second square (either in the same row or same column) are 1 or 3, because the absolute difference between 2 and any other number is at most 1. Suppose we put a 1 in the second square, then we can only put 3 in the third square (either in the same row or same column) because again the absolute difference between 1 and 3 is 2, which is more than 1. And we can keep going like this to fill in all the squares.
Now, when we have filled in all the squares, we can multiply all the numbers on the diagonal going from top left to bottom right, and also all the numbers on the diagonal going from top right to bottom left. The product we get is called the determinant.
Hadamard's Maximal Determinant Problem asks us to find the biggest determinant we can make using this method. It turns out that if we use a checkerboard pattern with some special numbers (1, -1 or 0), we can make the biggest determinant possible for any given size of checkerboard. This has important applications in areas like communication theory and coding theory.
A square pattern is like a checkerboard, with the same number of rows and columns. Imagine a 3x3 checkerboard with 9 squares. Now, what if we put a number in each square so that no two numbers are the same and the absolute value of the difference between any two numbers is one or less? This means that if we subtract one of the numbers from another number, the answer will be no bigger than one.
For example, let's say we put a 2 in the top left square. The only numbers we can put in the second square (either in the same row or same column) are 1 or 3, because the absolute difference between 2 and any other number is at most 1. Suppose we put a 1 in the second square, then we can only put 3 in the third square (either in the same row or same column) because again the absolute difference between 1 and 3 is 2, which is more than 1. And we can keep going like this to fill in all the squares.
Now, when we have filled in all the squares, we can multiply all the numbers on the diagonal going from top left to bottom right, and also all the numbers on the diagonal going from top right to bottom left. The product we get is called the determinant.
Hadamard's Maximal Determinant Problem asks us to find the biggest determinant we can make using this method. It turns out that if we use a checkerboard pattern with some special numbers (1, -1 or 0), we can make the biggest determinant possible for any given size of checkerboard. This has important applications in areas like communication theory and coding theory.