Number of Components in a Second Order System

We can write second order systems as a matrix. Any second order system can be written  
\[\mathbf{a}_{ij}\]
  or as a matrix.
\[ \left( \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} \right) \]

In general the  
\[\mathbf{a}_{ij}\]
  are not related so there are 3{sup}3{/sup =9 components. If the i,j can run from 1 to  
\[n\]
  there are  
\[n^2\]
  components. For a symmetric system  
\[\mathbf{a}_{ij} = a_{ji}\]

The matrix above becomes  
\[ \left( \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{12} & a_{22} & a_{23} \\ a_{13} & a_{23} & a_{33} \end{array} \right) \]

There are  
\[\frac{3(3-1)}{2} +3=6\]
  components.
If the i,j can run from 1 to  
\[n\]
  there are  
\[\frac{n(n-1)}{2} +n=\frac{n(n+1)}{2}\]
  components. For a skew symmetric system  
\[\mathbf{a}_{ij} = -a_{ji}\]

All the diagonal elements must be zero.
The matrix above becomes  
\[ \left( \begin{array}{ccc} 0 & a_{12} & a_{13} \\ -a_{12} & 0 & a_{23} \\ -a_{13} & -a_{23} & 0 \end{array} \right) \]

There are  
\[\frac{3(3-1)}{2}=3\]
  components.
If the i,j can run from 1 to  
\[n\]
  there are  
\[\frac{n(n-1)}{2}\]
  components.

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