For a matrix A, the tril(A) command returns a matrix R where R[i,j] = A[i,j] when R[i,j] is on or below the main diagonal of R. R[i,j] = 0, otherwise.

For a matrix A, the tril(A,k) command returns a matrix R where R[i,j] = A[i,j] when R[i,j] is on or below the diagonal of R indexed by k. R[i,j] = 0 otherwise.

The diagonals of a matrix are indexed using signed integers, where the main diagonal has index 0. Superdiagonals are indexed with positive integers and subdiagonals are indexed with negative integers.

$\mathrm{with}\left(\mathrm{MTM}\right)\:$

$A≔\mathrm{Matrix}\left(\left[\left[1\,2\,1\right]\,\left[4\,5\,6\right]\,\left[2\,8\,1\right]\right]\right)$

$A≔\left[\begin{array}{ccc}1& 2& 1\\ 4& 5& 6\\ 2& 8& 1\end{array}\right]$

$\mathrm{tril}\left(A\right)$

$\left[\begin{array}{ccc}1& 0& 0\\ 4& 5& 0\\ 2& 8& 1\end{array}\right]$

$\mathrm{tril}\left(A\,1\right)$

$\left[\begin{array}{ccc}1& 2& 0\\ 4& 5& 6\\ 2& 8& 1\end{array}\right]$

$\mathrm{tril}\left(A\,-1\right)$

$\left[\begin{array}{ccc}0& 0& 0\\ 4& 0& 0\\ 2& 8& 0\end{array}\right]$