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| Orthogonality and its
applications |
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| The Kronecker
delta function has many uses in mathematics; among the most important is
in linear algebra to express the inverse of an n×n matrix
with non-zero determinant. If we let Pn be
the lower triangular n×n matrix defined by
entries from Pascal's Triangle, the determinant is the product of the
main diagonal entries, which are all 1's, so we have a non-zero
determinant and the matrix has an inverse; Pn-1 is
a lower Pascal's triangular matrix with an alternating checkerboard
pattern of positive and negative entries where the main diagonal is all
positive. Here are a few examples for small n. |
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It's easy to see why the dot product of row j of Pn with column j of Pn-1 is always 1, since every pair of entries except entry j is either 0 in the row or 0 in the column, and the product at entry j is 1×1 = 1. For other entries, it is not quite as obvious that the alternating sum of the products will be 0. With the Java applet below, you can look at the alternating dot products in P15. Use the slider bars to select a row and column, and the non-zero products are shown at the bottom. positive in black and negative in red. The rows and columns are numbered 0 through 14, which makes the matrix position agree with the binomial coefficient numbers. If the boxes below are just grey rectangles, you can download Java for free through this link. |
applied
to each of the terms.
. The applet
below lets you change n, j andk, the row of the first matrix, the
column of the second matrix and the multiple which determines which rows
of Pascal's Triangle are being used in the second matrix, respectively.
