Pascal matrices

Every polynomial of degree n has n roots; every continuous function on [0, 1] attains its maximum; every real symmetric matrix has a complete set of orthonormal eigenvectors. “General theorems ” are a big part of the mathematics we know. We can hardly resist the urge to generalize further! Remove hypotheses, make the theorem tighter and more difficult, include more functions, move into Hilbert space,... It’s in our nature. The other extreme in mathematics might be called the “particular case”. One specific function or group or matrix becomes special. It obeys the general rules, like everyone else. At the same time it has some little twist that connects familiar objects in a neat way. This paper is about an extremely particular case. The familiar object is Pascal’s triangle. The little twist begins by putting that triangle of binomial coefficients into a matrix. Three different matrices—symmetric, lower triangular, and upper triangular—can hold Pascal’s triangle in a convenient way. Truncation produces n by n matrices Sn and Ln and Un—the pattern is visible for n = 4: ⎤⎡⎤⎡⎤