# Symmetric Matrix

A Symmetric Matrix is a square matrix, [math]\displaystyle{ A=[a_{ij}]_{\substack{i=1\dots m\\j=1\dots m}} }[/math], that remains unchanged by the matrix transpose operation ([math]\displaystyle{ \mathbf{A}^{\mathrm{T}} = \mathbf{A} }[/math]), that is if [math]\displaystyle{ a_{ij} = \begin{cases} a_{ji} & \quad \text{if } i\#j\\ a_{ij} & \quad \text{if } i=j\\ \end{cases} }[/math]

**Context:**- It can range from being a Binary Symmetric Matrix to being an Integer Symmetric Matrix to being a Real Symmetric Matrix to being a Complex Symmetric Matrix.

**Example(s):**- [math]\displaystyle{ \begin{bmatrix}1 & 2 & \infty \\ 2 & 4 & \pi \\ \infty & \pi & 6 \end{bmatrix} }[/math] because [math]\displaystyle{ a_{12}=a_{21}, a_{13}=a_{31}, a_{23}=a_{32} }[/math]
- a Co-Occurrence Matrix.
- a Covariance Matrix.
- a Symmetric Adjacency Matrix (representing an undirected graph).
- …

**Counter-Example(s):****See:**Off-Diagonal Element, Hermitian Matrix, Affinity Matrix, Conjugate Transpose, Main Diagonal, Diagonal Matrix, Self-Adjoint Operator.

## References

### 2015

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/symmetric_matrix Retrieved:2015-2-20.
- In linear algebra, a
**symmetric matrix**is a square matrix that is equal to its transpose. Formally, matrix*A*is symmetric if : [math]\displaystyle{ A = A^{\top}. }[/math] 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 = (

*a*_{ij}), then*a*_{ij}= a_{ji}, for all indices*i*and*j*.The following 3×3 matrix is symmetric: : [math]\displaystyle{ \begin{bmatrix} 1 & 7 & 3\\ 7 & 4 & -5\\ 3 & -5 & 6\end{bmatrix}. }[/math] Every square diagonal matrix is symmetric, since all off-diagonal entries are zero. Similarly, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.

In linear algebra, a real symmetric matrix represents a self-adjoint operator over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries. Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodations for them.

- In linear algebra, a