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):
  • a Skew-Symmetric Matrix.
  • a Nonsymmetric Matrix.
  • a Non-Square Matrix.
• See: Off-Diagonal Element, Hermitian Matrix, Affinity Matrix, Conjugate Transpose, Main Diagonal, Diagonal Matrix, Self-Adjoint Operator.

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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.

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