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Orthogonality and its applications



   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; better yet, the inverse must be another matrix with all integer entries, since every entries is the determinant of a sub-matrix divided by det(Pn) = 1.  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.



    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.


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        The patterns of the non-zero alternating dot product terms that add to zero turn out to be some binomial coefficient multiplied by an alternating row of Pascal's Triangle, which will be zero if the row has more than one entry.  This can be proven by applying the trinomial revision identity, which is shown below, to all the terms in the summation.

Trinomial revision: