lower triangular matrix

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:

• triangular matrix
• strictly triangular matrix
• upper triangular matrix