Symmetric matrix

A symmetric matrix over a field satisfies $A^T=A$. Its entries obey $a_{ij}=a_{ji}$. For example, $\left(\begin{array}{cc}2& 1\\ 1& 3\end{array}\right)$ is symmetric. Over $\mathbb{R}$, it represents a self-adjoint operator for the standard Euclidean inner product and has an orthonormal eigenbasis.

Over $\mathbb{C}$, symmetry must be distinguished from being Hermitian: transposition does not conjugate entries. The complex symmetric matrix $\left(\begin{array}{cc}i& 1\\ 1& -i\end{array}\right)$ is nonzero but squares to zero, so it is not diagonalizable. The real spectral theorem therefore cannot be extended to complex symmetric matrices merely by changing the coefficient field.

A symmetric matrix is a square matrix that is equal to its transpose. In mathematical terms, a matrix \( A \) is considered symmetric if: \[ A = A^T \] where \( A^T \) denotes the transpose of the matrix \( A \).