Symmetric Matrices

Abstract

In the 2-dimensional case, we saw that a special role is played by matrices $\left( {\begin{array}{*{20}{c}} a&b\\ b&d \end{array}} \right)$ which have both off-diagonal elements equal. The corresponding condition in 3 dimensions is symmetry about the diagonal. We say that a matrix is symmetric if the entry in the ith position in the jth column is the same as the entry in the jth position in the ith column, i.e., a _ij = a _ji for all i,j,

$$ \left( {\begin{array}{*{20}{c}} a&b&c\\ b&d&e\\ c&e&f \end{array}} \right)\; = \;\left( {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\ {{a_{12}}}&{{a_{22}}}&{{a_{23}}}\\ {{a_{13}}}&{{a_{23}}}&{{a_{33}}} \end{array}} \right). $$

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