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NewLinearAlgebraPaclet

Guides

  • Matrices

Symbols

  • AntidiagonallySymmetrizableMatrixQ
  • AntidiagonalMatrix
  • AntidiagonalMatrixQ
  • Antidiagonal
  • AntidiagonalTranspose
  • DesymmetrizedMatrix
  • DeTriangularizableMatrixQ
  • DeTriangularizeMatrix
  • HessianMatrix
  • JacobianMatrix
  • LeftArrowMatrix
  • LowerArrowMatrix
  • LowerRightTriangularize
  • LowerRightTriangularMatrixQ
  • MatrixSymmetrizability
  • PyramidMatrix
  • ReflectedDiagonalMatrix
  • RightArrowMatrix
  • TopArrowMatrix
  • UlamMatrix
  • UpperLeftTriangularize
  • UpperLeftTriangularMatrixQ
PeterBurbery`NewLinearAlgebraPaclet`
MatrixSymmetrizability
​
MatrixSymmetrizability
[matrix]
returns 1 if matrix is completely symmetric and 0 if matrix has no symmetry other than on the main diagonal. The closer the value is to 1, the more symmetralizable the matrix is.
​
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]:=
MatrixFormmatrixThatIsCompletelySymmetric=
PyramidMatrix
[18]
Out[1]//MatrixForm=
1
1
1
1
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1
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In[2]:=
SymmetricMatrixQ[matrixThatIsCompletelySymmetric]
Out[2]=
True
In[3]:=
MatrixSymmetrizability
[matrixThatIsCompletelySymmetric]
Out[3]=
1
In[4]:=
MatrixFormmatrixWithAHighDegreeOfSymmetry=
matrix example

Out[4]//MatrixForm=
1
1
1
1
1
1
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1
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1
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1
1
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2
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In[5]:=
MatrixSymmetrizability
[matrixWithAHighDegreeOfSymmetry]
Out[5]=
152
153
In[6]:=
N
MatrixSymmetrizability
[matrixWithAHighDegreeOfSymmetry]
Out[6]=
0.993464
In[7]:=
MatrixFormmatrixWithALowDegreeOfSymmetry=
1
4
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32
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1
1
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2
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
Out[7]//MatrixForm=
1
4
5
6
13
21
32
3
1
1
7
14
22
33
2
1
1
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15
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34
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In[8]:=
MatrixSymmetrizability
[matrixWithALowDegreeOfSymmetry]
Out[8]=
2
21
In[9]:=
N
MatrixSymmetrizability
[matrixWithALowDegreeOfSymmetry]
Out[9]=
0.0952381
Here are the symmetric parts of the matrix:
In[10]:=
MatrixFormReplaceAtmatrixWithALowDegreeOfSymmetry,x_Highlighted[x,StripOnInputTrue],PositionNormal
DesymmetrizedMatrix
[matrixWithALowDegreeOfSymmetry],0,{2},HeadsFalse
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
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16
24
35
17
14
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25
36
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37
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Here is a matrix that is completely not symmetrical:
In[11]:=
MatrixForm[notSymmetricalMatrix=Partition[Range[100],10]]
Out[11]//MatrixForm=
1
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In[12]:=
MatrixSymmetrizability
[notSymmetricalMatrix]
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.
SeeAlso
DesymmetrizedMatrix
 
▪
Symmetrize
 
▪
SymmetrizedArray
 
▪
Symmetric
 
▪
Antisymmetric
 
▪
SymmetricMatrixQ
 
▪
AntisymmetricMatrixQ
 
▪
Transpose
 
▪
TensorSymmetry
RelatedGuides
▪
Matrices
""

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