A matrix is upper triangular if all its entries below the diagonal are zero. A matrix is lower triangular if all its entries above the diagonal are zero. If, in addition, the diagonal is zero, then the matrix is strictly upper triangular and strictle lower triangular respectively.

A triangular matrix is either upper or lower triangular. A strictly triangular matrix is either strictly upper triangular or strictly lower triangular.

A unit triangular matrix is a triangular matrix (upper or lower) whose diagonal entries are one. Somtimes such matrices are called unitriangular (a unitriangular matrix). So we speak of lower unit triangular, upper unit triangular, lower unitriangular and lower unitriangular matrices.

Some authors call lower triangular matrices left triangular and upper triangular matrices right triangular. Historically, some authors have called triangular matrices semidiagonal.