A lower triangular matrix is a square matrix for which every value above the diagonal must be 0. That is to say:

[ a1,1 0 0 ... 0 ]
[ a2,1 a2,2 0 ... 0 ]
A = [ a3,1 a3,2 a3,3 ... 0 ]
[ ... ... ... ... ... ]
[ an,1 an,2 an,3 ... an,n ]

Also defined is the *strictly lower triangular matrix* for which values on the diagonal must also be 0:

[ 0 0 0 ... 0 0 ]
[ a2,1 0 0 ... 0 0 ]
A = [ a3,1 a3,2 0 ... 0 0 ]
[ ... ... ... ... ... ... ]
[ an,1 an,2 an,3 ... an,n-1 0 ]

A more formal definition for the standard lower triangular matrix would be

a_{i,j} =

{ a_{i,j}, `i` ≥ `j`

{ 0, `i` < `j`

And for the strictly lower triangular matrix:

a_{i,j} =

{ a_{i,j}, `i` > `j`

{ 0, `i` ≤ `j`

Also see: