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Tricks of the Trade
Pascal MatricesWriting Pascal's triangle as a lower triangular matrix generates one type of Pascal matrix (see mathworld.wolfram.com/PascalMatrix.html). Such matrices have many interesting properties (web.mit.edu/18.06/www/pascalwork.pdf). Define the lower triangular matrix by , where .
Here is .
And here is its inverse.
In general, the inverse of has Clearly .
Define .
Using the binomial theorem, we can show that , that is,
"Prove" the binomial theorem.
Verify that for .
Writing , it follows that . Here is .
In general, it can be shown that . Check that .
Define the upper triangular matrix by alternating the sign of the rows in .
Here is .
is its own inverse (it is involutory).
Multiplying by its transpose yields a symmetric, positivedefinite matrix, denoted , now with Pascal's triangle entries along each skew diagonal (see mathworld.wolfram.com/SkewDiagonal.html).
Since and , we obtain that .
The inverse of has integer entries.
It can be shown that . That is is similar to (see mathworld.wolfram.com/SimilarMatrices.html). Verify that for .
Since similar matrices have the same eigenvalues, the eigenvalues of must come in reciprocal pairs, and . Moreover, if is odd, then the "middle" eigenvalue must be unity. Here are the eigenvalues of .
Confirm that the eigenvalues come in reciprocal pairs.


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