Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
PeterBurbery`NewLinearAlgebraPaclet`
MatrixSymmetrizability  |
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Details and Options
Examples
(1)
Basic Examples
(1)
We will look at a matrix that is completely symmetric, a matrix with a high degree of symmetry, a matrix with a low degree of symmetry, and a matrix with no symmetry at all.
In[1]:=
MatrixFormmatrixThatIsCompletelySymmetric=
[18]
 | PyramidMatrix |
Out[1]//MatrixForm=
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 |
1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 5 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 7 | 7 | 7 | 7 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 8 | 8 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 8 | 8 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 7 | 7 | 7 | 7 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 5 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 2 | 1 |
1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
In[2]:=
SymmetricMatrixQ[matrixThatIsCompletelySymmetric]
Out[2]=
True
In[3]:=
 | MatrixSymmetrizability |
Out[3]=
1
In[4]:=
MatrixFormmatrixWithAHighDegreeOfSymmetry=
matrix example |
Out[4]//MatrixForm=
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 |
1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 5 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 7 | 7 | 7 | 7 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 8 | 8 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 8 | 8 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 7 | 7 | 7 | 7 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 5 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 4 | 4 | 2 | 1 |
1 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 3 | 2 | 1 |
1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 2 | 1 |
1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
In[5]:=
| MatrixSymmetrizability |
Out[5]=
152
153
In[6]:=
N
[matrixWithAHighDegreeOfSymmetry]
 | MatrixSymmetrizability |
Out[6]=
0.993464
In[7]:=
MatrixFormmatrixWithALowDegreeOfSymmetry=
1 | 4 | 5 | 6 | 13 | 21 | 32 |
3 | 1 | 1 | 7 | 14 | 22 | 33 |
2 | 1 | 1 | 8 | 15 | 23 | 34 |
9 | 10 | 11 | 12 | 16 | 24 | 35 |
17 | 14 | 18 | 19 | 20 | 25 | 36 |
26 | 27 | 28 | 29 | 30 | 31 | 37 |
38 | 39 | 40 | 41 | 42 | 43 | 44 |
Out[7]//MatrixForm=
1 | 4 | 5 | 6 | 13 | 21 | 32 |
3 | 1 | 1 | 7 | 14 | 22 | 33 |
2 | 1 | 1 | 8 | 15 | 23 | 34 |
9 | 10 | 11 | 12 | 16 | 24 | 35 |
17 | 14 | 18 | 19 | 20 | 25 | 36 |
26 | 27 | 28 | 29 | 30 | 31 | 37 |
38 | 39 | 40 | 41 | 42 | 43 | 44 |
In[8]:=
| MatrixSymmetrizability |
Out[8]=
2
21
In[9]:=
N
[matrixWithALowDegreeOfSymmetry]
| MatrixSymmetrizability |
Out[9]=
0.0952381
Here are the symmetric parts of the matrix:
In[10]:=
MatrixFormReplaceAtmatrixWithALowDegreeOfSymmetry,x_Highlighted[x,StripOnInputTrue],PositionNormal
[matrixWithALowDegreeOfSymmetry],0,{2},HeadsFalse
 | DesymmetrizedMatrix |
Out[10]//MatrixForm=
1 | 4 | 5 | 6 | 13 | 21 | 32 |
3 | 1 | 1 | 7 | 14 | 22 | 33 |
2 | 1 | 1 | 8 | 15 | 23 | 34 |
9 | 10 | 11 | 12 | 16 | 24 | 35 |
17 | 14 | 18 | 19 | 20 | 25 | 36 |
26 | 27 | 28 | 29 | 30 | 31 | 37 |
38 | 39 | 40 | 41 | 42 | 43 | 44 |
Here is a matrix that is completely not symmetrical:
In[11]:=
MatrixForm[notSymmetricalMatrix=Partition[Range[100],10]]
Out[11]//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 | 50 |
51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |
61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 |
71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 |
81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |
91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |
In[12]:=
| MatrixSymmetrizability |
Out[12]=
0
When the matrix symmetrizability is 0, it does not mean that the matrix cannot be symmetrized. It just means that there are no symmetric elements, other than the diagonal.
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