In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
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 a i j {\displaystyle a_{ij}} denotes the entry in the i {\displaystyle i} th row and j {\displaystyle j} th column then
Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2 {\displaystyle 2} , 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 represented in an orthonormal basis 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.
Example The following 3 × 3 {\displaystyle 3\times 3} matrix is symmetric:
A = [ 1 7 3 7 4 5 3 5 2 ] ; {\displaystyle A={\begin{bmatrix}1&7&3\\7&4&5\\3&5&2\end{bmatrix}};}
indeed, A = A T {\displaystyle A=A^{\textsf {T}}} .
Properties
Basic properties The sum and difference of two symmetric matrices are symmetric. This is not always true for the product: given symmetric matrices A {\displaystyle A} and B {\displaystyle B} , the product A B {\displaystyle AB} is symmetric if and only if A {\displaystyle A} and B {\displaystyle B} commute, i.e., A B = B A {\displaystyle AB=BA} . If A {\displaystyle A} is symmetric, then for any integer k ≥ 0 {\displaystyle k\geq 0} , {\displaystyle \,}
A k {\displaystyle A^{k}} is also symmetric. If A − 1 {\displaystyle A^{-1}} exists, it is symmetric if and only if A {\displaystyle A} is symmetric. The rank of a symmetric matrix A {\displaystyle A} is equal to the number of non-zero eigenvalues of A {\displaystyle A} .
Decomposition into symmetric and skew-symmetric Any square matrix can uniquely be written as sum of a symmetric and a skew-symmetric matrix. This decomposition is known as the Toeplitz decomposition. Let Mat n {\displaystyle {\mbox{Mat}}_{n}} denote the space of n × n {\displaystyle n\times n} matrices. If Sym n {\displaystyle {\mbox{Sym}}_{n}} denotes the space of n × n {\displaystyle n\times n} symmetric matrices and Skew n {\displaystyle {\mbox{Skew}}_{n}} the space of n × n {\displaystyle n\times n} skew-symmetric matrices, then
Mat n = Sym n + Skew n and Sym n ∩ Skew n = { 0 } , {\displaystyle {\mbox{Mat}}_{n}={\mbox{Sym}}_{n}+{\mbox{Skew}}_{n}\qquad {\mbox{and}}\qquad {\mbox{Sym}}_{n}\cap {\mbox{Skew}}_{n}=\{0\},}
i.e.
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