In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix A is symmetric if
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if the entries are written as A = (aij), then aij = aji, for all indices i and j.
![A = A^{\textrm{T}} , \,\!](/Images/godic/202502/14/71e2036b60520bda9b45a26b0f37a4c32355.png")
![A = (a_{ij})](/Images/godic/202502/14/32731aa33f64bd1543021cb1abc139332355.png")
，则：![a_{ij} = a_{ji} \,\!](/Images/godic/202502/14/ad078a5fa1739ecc77d3f96d1111fc112355.png")
![\begin{bmatrix}
1 & 2 & 3\\
2 & 4 & -5\\
3 & -5 & 6\end{bmatrix}](/Images/godic/202502/14/ad0085d01b1b9c0ae0471b4e0d138d842355.png")
![\begin{bmatrix}
0 & -3 & 4\\
3 & 0 & -5\\
-4 & 5 & 0\end{bmatrix}](/Images/godic/202502/14/bd068b4c13e6972b5185a68125454aa12355.png")