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m Symmetric matrices represent self-adjoint operators only if the basis is orthonormal. |
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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.
== Example ==
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