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Pascal Matrices

Writing 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/pascal-work.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, positive-definite 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.