Triangular Matrix

An upper triangular Matrix ${\hbox{\sf U}}$ is defined by

$$U_{ij} = \cases{ a_{ij} & for $i \leq j$\cr 0 & for $i > j$.\cr}$$ (1)

Written explicitly,

$${\hbox{\sf U}} = \left[{\matrix{ a_{11} & a_{12} & \cdots &... ...ots & \ddots & \vdots\cr 0 & 0 & \cdots & a_{nn}\cr}}\right].$$ (2)

A lower triangular Matrix ${\hbox{\sf L}}$ is defined by

$$L_{ij} = \cases{ a_{ij} & for $i \geq j$\cr 0 & for $i < j$.\cr}$$ (3)

Written explicitly,

$${\hbox{\sf L}} = \left[{\matrix{ a_{11} & 0 & \cdots & 0\cr... ... \ddots & 0\cr a_{n1} & a_{n2} & \cdots & a_{nn}\cr}}\right].$$ (4)

See also Hessenberg Matrix, Hilbert Matrix, Matrix, Vandermonde Matrix