# Symmetric matrix

## Matrix equal to its transpose / From Wikipedia, the free encyclopedia

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In linear algebra, a **symmetric matrix** is a square matrix that is equal to its transpose. Formally,

$A{\text{ is symmetric}}\iff A=A^{\textsf {T}}.$

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 $a_{ij}$ denotes the entry in the $i$th row and $j$th column then

$A{\text{ is symmetric}}\iff {\text{ for every }}i,j,\quad a_{ji}=a_{ij}$

for all indices $i$ and $j.$

Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 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^{[1]} 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.