Symmetric matrix

In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, a symmetric matrix is a <a href="/facts/Square_matrix/5TP9mNNn">square matrix</a> that is equal to its <a href="/facts/Transpose/8wmsagGS">transpose</a>. Formally,

A
 
  is symmetric
 
 
 ⟺
 
 A
 =
 
 A
 
 
 T
 
 
 
 .
 
 
 {\displaystyle 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 <a href="/facts/Main_diagonal/rkT2F08Q">main diagonal</a>. 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

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

for all indices 
 
 
 
 i
 
 
 {\displaystyle i}
 
 and 
 
 
 
 j
 .
 
 
 {\displaystyle j.}

Every square <a href="/facts/Diagonal_matrix/tjIAHrE6">diagonal matrix</a> is symmetric, since all off-diagonal elements are zero. Similarly in <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> different from 2, each diagonal element of a <a href="/facts/Skew-symmetric_matrix/nw5CQeLN">skew-symmetric matrix</a> must be zero, since each is its own negative.
In linear algebra, a <a href="/facts/Real_number/R02gw5Pb">real</a> symmetric matrix represents a <a href="/facts/Self-adjoint_operator/VrH1nEAK">self-adjoint operator</a> represented in an <a href="/facts/Orthonormal_basis/pzfKHlJG">orthonormal basis</a> over a <a href="/facts/Real_number/R02gw5Pb">real</a> <a href="/facts/Inner_product_space/HoyjElEy">inner product space</a>. The corresponding object for a <a href="/facts/Complex_number/2w7mqI7e">complex</a> inner product space is a <a href="/facts/Hermitian_matrix/qArPblYo">Hermitian matrix</a> with complex-valued entries, which is equal to its <a href="/facts/Conjugate_transpose/7eBWtqVG">conjugate transpose</a>. 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.

Symmetric matrix open-in-new

Symmetric matrix