View source for Symmetric matrix

A square matrix in which any two elements symmetrically positioned with respect to the main diagonal are equal to each other, that is, a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s0916801.png" /> that is equal to its transpose:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s0916802.png" /></td> </tr></table>

A real symmetric matrix of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s0916803.png" /> has exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s0916804.png" /> real eigenvalues (counted with multiplicity). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s0916805.png" /> is a symmetric matrix, then so are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s0916806.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s0916807.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s0916808.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s0916809.png" /> are symmetric matrices of the same order, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s09168010.png" /> is a symmetric matrix, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s09168011.png" /> is symmetric if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s09168012.png" />.



====Comments====
Every square complex matrix is similar to a symmetric matrix. A real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s09168013.png" />-matrix is symmetric if and only if the associated operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s09168014.png" /> (with respect to the standard basis) is self-adjoint (with respect to the standard inner product). A [[Polar decomposition|polar decomposition]] factors a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s09168015.png" /> into a product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s09168016.png" /> of a symmetric and an orthogonal matrix.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s09168017.png" /> be a bilinear form on a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s09168018.png" /> (cf. [[Bilinear mapping|Bilinear mapping]]). Then the matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s09168019.png" /> (with respect to the same basis in the two factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s09168020.png" />) is symmetric if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s09168021.png" /> is a symmetric bilinear form, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091680/s09168022.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,   "The theory of matrices" , '''1''' , Chelsea, reprint  (1959–1960)  pp. Vol. 1, Chapt. IX; Vol. 2, Chapt. XI  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Noll,   "Finite dimensional spaces" , M. Nijhoff  (1987)  pp. Sect. 2.7</TD></TR></table>