# Definition of symmetric matrix

Put content here.**Definition.** A matrix $A$ is *symmetric* if it equals its transpose: $A = A^T$.
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Equivalently, $a_{ij} = a_{ji}$ for all $i, j$. A symmetric matrix is always square.
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**Example.** $A = \begin{pmatrix} 1 & 3 & 0 \\ 3 & -2 & 4 \\ 0 & 4 & 5 \end{pmatrix}$ is symmetric.
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**Properties:**
- The sum of symmetric matrices is symmetric
- If $A$ and $B$ commute, then $AB$ is symmetric
- For any matrix $M$, both $MM^T$ and $M^T M$ are symmetric
- The diagonal entries of a symmetric matrix can be arbitrary, but the matrix is determined by $n(n+1)/2$ independent entries

# Parents

* Symmetric matrices