Symmetric Matrices

A symmetric matrix is a square matrix that is equal to its own transpose. In other words, it’s a mirror image of itself across the main diagonal (from top-left to bottom-right).

\[A = A^T\]

Example

A matrix is symmetric if the element in the \(i\)-th row and \(j\)-th column is the same as the element in the \(j\)-th row and \(i\)-th column (\(A_{ij} = A_{ji}\)).

\[\begin{bmatrix} A_{11} & \color{blue}{A_{12}} & \color{green}{A_{13}} \\ \color{blue}{A_{21}} & A_{22} & \color{purple}{A_{23}} \\ \color{green}{A_{31}} & \color{purple}{A_{32}} & A_{33} \end{bmatrix} = \begin{bmatrix} 1 & \color{blue}2 & \color{green}3 \\ \color{blue}2 & 4 & \color{purple}5 \\ \color{green}3 & \color{purple}5 & 6 \end{bmatrix}\]