Editing Symmetric matrix (section)

{{Short description|Matrix equal to its transpose
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{{about|a matrix symmetric about its diagonal|a matrix symmetric about its center| Centrosymmetric matrix}}
{{For|matrices with symmetry over the [[complex number]] field|Hermitian matrix}}
{{Use American English|date=January 2019}}
[[File:Matrix symmetry qtl1.svg|thumb|Symmetry of a 5×5 matrix]]

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

{{Equation box 1
|indent =:
|equation = <math>A \text{ is symmetric} \iff A = A^\textsf{T}.</math>
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|border colour = #0073CF
|background colour = #F5FFFA
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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 <math>a_{ij}</math> denotes the entry in the <math>i</math>th row and <math>j</math>th column then

{{Equation box 1
|indent = :
|equation = <math>A \text{ is symmetric} \iff \text{ for every }i,j,\quad a_{ji} = a_{ij}</math>
|cellpadding= 6
|border colour = #0073CF
|background colour = #F5FFFA
}}

for all indices <math>i</math> and <math>j.</math>

Every square [[diagonal matrix]] is symmetric, since all off-diagonal elements are zero. Similarly in [[characteristic (algebra)|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 number|real]] symmetric matrix represents a [[self-adjoint operator]]<ref>{{Cite book|author=Jesús Rojo García|title=Álgebra lineal |language= es|edition=2nd|publisher=Editorial AC|year=1986|isbn=84-7288-120-2}}</ref> represented in an [[orthonormal basis]] over a [[real number|real]] [[inner product space]]. The corresponding object for a [[complex number|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.