[Numeric-interest] Name for matrix with "nice" structure for LU factorization?

[Douglas Priest]

Hello,

Is there a name associated with the matrix A(a) = LU(a), where
L is unit lower-triangular with all 1's below the diagonal and
U(a) is upper-triangular with constant a on the diagonal and all
1's above the diagonal?

Any related background information, such as a bibliographic
citation or interesting anecdotes, would be appreciated.

Thank you,
Doug Priest