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linalg::pascal(n) returns the n×n Pascal
matrix P given
≤ i, j ≤ n.
The entries of Pascal matrices are integer numbers. Note, however,
that the returned matrix is not defined over the component domain
but over the standard component domain
Dom::ExpressionField(). Thus, no conversion is necessary when working with other
functions that expect or return matrices over that component domain.
linalg::pascal(n, Dom::Integer) to define
the n×n Pascal
matrix over the ring of integer numbers.
Inverse Pascal matrices are provided by
We construct the 3×3 Pascal matrix:
This is a matrix of the domain
If you prefer a different component ring, the matrix may be
converted to the desired domain after construction (see
coerce, for example).
Alternatively, one can specify the component ring when creating the
Pascal matrix. For example, specification of the domain
The Cholesky factor of a Pascal matrix consists of the elements of Pascal's triangle:
The dimension of the matrix: a positive integer
of the domain
Pascal matrices are symmetric and positive definite.
The determinant of a Pascal matrix is 1.
The inverse of a Pascal matrix has integer entries.
If λ is an eigenvalue of a Pascal matrix, then is also an eigenvalue of the matrix.