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Unformatted text preview: Pascal Matrices Alan Edelman and Gilbert Strang Department of Mathematics, Massachusetts Institute of Technology 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 dicult, include more functions, move into Hilbert space,. . . Its 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 Pascals triangle . The little twist begins by putting that triangle of binomial coecients into a matrix. Three different matrices symmetric, lower triangular , and upper triangular can hold Pascals triangle in a convenient way. Truncation produces n by n matrices S n and L n and U n the pattern is visible for n = 4: 1 1 1 1 1 1 1 1 1 S 4 = 1 2 3 4 1 3 6 10 L 4 = 1 1 1 2 1 U 4 = 1 2 3 1 3 . 1 4 10 20 1 3 3 1 1 We mention first a very specific fact: The determinant of every S n is 1. (If we emphasized det L n = 1 and det U n = 1, you would write to the Editor. Too special !) Determinants are often a surface reection of a deeper property within the matrix. That is true here, and the connection between the three matrices is quickly revealed. It holds for every n : S equals L times U and then (det S ) = (det L )(det U ) = 1 . This identity S = LU is an instance of one of the four great matrix factorizations of linear algebra : 1 1. Triangular times triangular: A = LU from Gaussian elimination 2. Orthogonal times triangular: A = QR from Gram-Schmidt 3. Orthogonal times diagonal times orthogonal: A = U V T with the singular values in 4. Diagonalization: A = S S 1 with eigenvalues in and eigenvectors in S . Symmetric matrices allow S 1 = S T orthonormal eigenvectors and real eigenvalues in the spectral theorem. In A = LU , the triangular U is the goal of elimination. The pivots lie on its diagonal (those are ratios det A n / det A n 1 , so the pivots for Pascal are all 1s). We reach U by row operations that are recorded in L . Then A x = b is solved by forward elimination and back substitution. In principle this is straightforward, but the cost adds up: billions a year for the most frequently used algorithm in scientific computing....
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