upper triangular matrix

A matrix of the form:

|* * *|
|0 * *|
|0 0 *|

Where all numbers lying below the main diagonal are zero.

An upper triangular matrix is a square matrix for which every value below the diagonal must be 0. That is to say:

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

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

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

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

a_i,j =

{ 0, i > j

{ a_i,j, i ≤ j

And for the strictly upper triangular matrix:

a_i,j =

{ 0, i ≥ j

{ a_i,j, i < j

Also see:

• triangular matrix
• strictly triangular matrix
• lower triangular matrix