Symmetric tensor

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a symmetric tensor is an <a href="/facts/Mixed_tensor/I5H8r3a2">unmixed</a> <a href="/facts/Tensor/pNjFo5tV">tensor</a> that is invariant under a <a href="/facts/Permutation/JSw9ukmv">permutation</a> of its vector arguments:

T
        (
        
          v
          
            1
          
        
        ,
        
          v
          
            2
          
        
        ,
        …
        ,
        
          v
          
            r
          
        
        )
        =
        T
        (
        
          v
          
            σ
            1
          
        
        ,
        
          v
          
            σ
            2
          
        
        ,
        …
        ,
        
          v
          
            σ
            r
          
        
        )
      
    
    {\displaystyle T(v_{1},v_{2},\ldots ,v_{r})=T(v_{\sigma 1},v_{\sigma 2},\ldots ,v_{\sigma r})}

for every permutation σ of the symbols {1, 2, ..., r}. Alternatively, a symmetric tensor of order r represented in coordinates as a quantity with r indices satisfies

T
          
            
              i
              
                1
              
            
            
              i
              
                2
              
            
            ⋯
            
              i
              
                r
              
            
          
        
        =
        
          T
          
            
              i
              
                σ
                1
              
            
            
              i
              
                σ
                2
              
            
            ⋯
            
              i
              
                σ
                r
              
            
          
        
        .
      
    
    {\displaystyle T_{i_{1}i_{2}\cdots i_{r}}=T_{i_{\sigma 1}i_{\sigma 2}\cdots i_{\sigma r}}.}

The space of symmetric tensors of order r on a finite-dimensional <a href="/facts/Vector_space/M1pxMLx2">vector space</a> V is <a href="/facts/Natural_isomorphism/xwPumTHt">naturally isomorphic</a> to the dual of the space of <a href="/facts/Homogeneous_polynomial/WUHL7Kct">homogeneous polynomials</a> of degree r on V. Over <a href="/facts/Field_(mathematics)/xAjAS4ko">fields</a> of <a href="/facts/Characteristic_zero/D2VzcQaG">characteristic zero</a>, the <a href="/facts/Graded_vector_space/Ff1maywV">graded vector space</a> of all symmetric tensors can be naturally identified with the <a href="/facts/Symmetric_algebra/A8FthY0N">symmetric algebra</a> on V. A related concept is that of the <a href="/facts/Antisymmetric_tensor/atcSaPST">antisymmetric tensor</a> or <a href="/facts/Alternating_form/rdN4TvpR">alternating form</a>. Symmetric tensors occur widely in <a href="/facts/Engineering/RFVg8H3I">engineering</a>, <a href="/facts/Physics/a7aH2y08">physics</a> and <a href="/facts/Mathematics/pxTouaz4">mathematics</a>.

Symmetric tensor open-in-new

Symmetric tensor