Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
PeterBurbery`NewLinearAlgebraPaclet`
DeTriangularizeMatrix  |
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Details and Options
Examples
(1)
Basic Examples
(1)
Detriangularize a lower-triangularized pyramid matrix:
To get a lower-triangularized or upper-triangularized matrix, use the functions and , respectively.
In[1]:=
MatrixFormpyramidMatrix=
[7]
 | PyramidMatrix |
Out[1]//MatrixForm=
1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 2 | 2 | 2 | 2 | 2 | 1 |
1 | 2 | 3 | 3 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 3 | 3 | 2 | 1 |
1 | 2 | 2 | 2 | 2 | 2 | 1 |
1 | 1 | 1 | 1 | 1 | 1 | 1 |
The matrix that you triangularize should be symmetric:
In[2]:=
SymmetricMatrixQ[pyramidMatrix]
Out[2]=
True
In[3]:=
MatrixForm[lowerTriangularMatrix=LowerTriangularize[pyramidMatrix]]
Out[3]//MatrixForm=
1 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 2 | 0 | 0 | 0 | 0 | 0 |
1 | 2 | 3 | 0 | 0 | 0 | 0 |
1 | 2 | 3 | 4 | 0 | 0 | 0 |
1 | 2 | 3 | 3 | 3 | 0 | 0 |
1 | 2 | 2 | 2 | 2 | 2 | 0 |
1 | 1 | 1 | 1 | 1 | 1 | 1 |
In[4]:=
MatrixForm
[LowerTriangularize[pyramidMatrix]]
 | DeTriangularizeMatrix |
Out[4]//MatrixForm=
1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 2 | 2 | 2 | 2 | 2 | 1 |
1 | 2 | 3 | 3 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 3 | 3 | 2 | 1 |
1 | 2 | 2 | 2 | 2 | 2 | 1 |
1 | 1 | 1 | 1 | 1 | 1 | 1 |
We have recovered the original matrix:
In[5]:=
pyramidMatrix===
[LowerTriangularize[pyramidMatrix]]
| DeTriangularizeMatrix |
Out[5]=
True
This would not have worked if the matrix was not symmetric:
In[6]:=
MatrixForm[nonSymmetricMatrix=Partition[Range[49],7]]
Out[6]//MatrixForm=
1 | 2 | 3 | 4 | 5 | 6 | 7 |
8 | 9 | 10 | 11 | 12 | 13 | 14 |
15 | 16 | 17 | 18 | 19 | 20 | 21 |
22 | 23 | 24 | 25 | 26 | 27 | 28 |
29 | 30 | 31 | 32 | 33 | 34 | 35 |
36 | 37 | 38 | 39 | 40 | 41 | 42 |
43 | 44 | 45 | 46 | 47 | 48 | 49 |
In[7]:=
SymmetricMatrixQ[nonSymmetricMatrix]
Out[7]=
False
In[8]:=
MatrixForm[lowerTriangularizedNonSymmetricMatrix=LowerTriangularize[nonSymmetricMatrix]]
Out[8]//MatrixForm=
1 | 0 | 0 | 0 | 0 | 0 | 0 |
8 | 9 | 0 | 0 | 0 | 0 | 0 |
15 | 16 | 17 | 0 | 0 | 0 | 0 |
22 | 23 | 24 | 25 | 0 | 0 | 0 |
29 | 30 | 31 | 32 | 33 | 0 | 0 |
36 | 37 | 38 | 39 | 40 | 41 | 0 |
43 | 44 | 45 | 46 | 47 | 48 | 49 |
In[9]:=
MatrixForm
[lowerTriangularizedNonSymmetricMatrix]
| DeTriangularizeMatrix |
Out[9]//MatrixForm=
1 | 8 | 15 | 22 | 29 | 36 | 43 |
8 | 9 | 16 | 23 | 30 | 37 | 44 |
15 | 16 | 17 | 24 | 31 | 38 | 45 |
22 | 23 | 24 | 25 | 32 | 39 | 46 |
29 | 30 | 31 | 32 | 33 | 40 | 47 |
36 | 37 | 38 | 39 | 40 | 41 | 48 |
43 | 44 | 45 | 46 | 47 | 48 | 49 |
In[10]:=
| DeTriangularizeMatrix |
Out[10]=
False
Here is an example of doing it with an upper triangular matrix:
In[11]:=
MatrixForm[upperTriangularMatrix=UpperTriangularize[pyramidMatrix]]
Out[11]//MatrixForm=
1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 2 | 2 | 2 | 2 | 2 | 1 |
0 | 0 | 3 | 3 | 3 | 2 | 1 |
0 | 0 | 0 | 4 | 3 | 2 | 1 |
0 | 0 | 0 | 0 | 3 | 2 | 1 |
0 | 0 | 0 | 0 | 0 | 2 | 1 |
0 | 0 | 0 | 0 | 0 | 0 | 1 |
In[12]:=
MatrixForm
[upperTriangularMatrix]
| DeTriangularizeMatrix |
Out[12]//MatrixForm=
1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 2 | 2 | 2 | 2 | 2 | 1 |
1 | 2 | 3 | 3 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 3 | 3 | 2 | 1 |
1 | 2 | 2 | 2 | 2 | 2 | 1 |
1 | 1 | 1 | 1 | 1 | 1 | 1 |
In[13]:=
| DeTriangularizeMatrix |
Out[13]=
True
Here is another example of it not working, but with an upper triangular matrix this time.
In[14]:=
MatrixForm[upperTriangularizedNonSymmetricMatrix=UpperTriangularize[nonSymmetricMatrix]]
Out[14]//MatrixForm=
1 | 2 | 3 | 4 | 5 | 6 | 7 |
0 | 9 | 10 | 11 | 12 | 13 | 14 |
0 | 0 | 17 | 18 | 19 | 20 | 21 |
0 | 0 | 0 | 25 | 26 | 27 | 28 |
0 | 0 | 0 | 0 | 33 | 34 | 35 |
0 | 0 | 0 | 0 | 0 | 41 | 42 |
0 | 0 | 0 | 0 | 0 | 0 | 49 |
In[15]:=
MatrixForm
[upperTriangularizedNonSymmetricMatrix]
| DeTriangularizeMatrix |
Out[15]//MatrixForm=
1 | 2 | 3 | 4 | 5 | 6 | 7 |
2 | 9 | 10 | 11 | 12 | 13 | 14 |
3 | 10 | 17 | 18 | 19 | 20 | 21 |
4 | 11 | 18 | 25 | 26 | 27 | 28 |
5 | 12 | 19 | 26 | 33 | 34 | 35 |
6 | 13 | 20 | 27 | 34 | 41 | 42 |
7 | 14 | 21 | 28 | 35 | 42 | 49 |
In[16]:=
| DeTriangularizeMatrix |
Out[16]=
False
The subdiagonals are below the main diagonal and the super diagonals are above the main diagonal.
The function will work if a symmetric matrix was upper triangularized on a superdiagonal or lower-triangularized on a subdiagonal. The function will not work if a symmetric matrix was upper triangularized on a subdiagonal or lower triangularized on a super diagonal.
In[17]:=
MatrixForm[upperTriangularizedOnASuperDiagonal=UpperTriangularize[pyramidMatrix,2]]
Out[17]//MatrixForm=
0 | 0 | 1 | 1 | 1 | 1 | 1 |
0 | 0 | 0 | 2 | 2 | 2 | 1 |
0 | 0 | 0 | 0 | 3 | 2 | 1 |
0 | 0 | 0 | 0 | 0 | 2 | 1 |
0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 |
In[18]:=
MatrixForm
[upperTriangularizedOnASuperDiagonal]
| DeTriangularizeMatrix |
Out[18]//MatrixForm=
0 | 0 | 1 | 1 | 1 | 1 | 1 |
0 | 0 | 0 | 2 | 2 | 2 | 1 |
1 | 0 | 0 | 0 | 3 | 2 | 1 |
1 | 2 | 0 | 0 | 0 | 2 | 1 |
1 | 2 | 3 | 0 | 0 | 0 | 1 |
1 | 2 | 2 | 2 | 0 | 0 | 0 |
1 | 1 | 1 | 1 | 1 | 0 | 0 |
In this case, we don't compare this to the original matrix because its different than the original matrix.
Let's look at a matrix that is upper-triangularized on a subdiagonal
Let's do the same thing but with a lower-triangularized matrix.