Editing Square matrix (section)

===Definite matrix===
{| class="wikitable" style="float:right; text-align:center; margin:0ex 0ex 2ex 2ex;"
|-
! [[Positive definite matrix|Positive definite]] !! [[Indefinite matrix|Indefinite]]
|-
| <math> \begin{bmatrix}
         1/4 & 0 \\
         0 & 1 \\
     \end{bmatrix} </math>
| <math> \begin{bmatrix}
         1/4 & 0 \\
         0 & -1/4 
     \end{bmatrix} </math>
|-
| {{math|1=''Q''(''x'',''y'') = 1/4 ''x''<sup>2</sup> + ''y''<sup>2</sup>}}
| {{math|1=''Q''(''x'',''y'') = 1/4 ''x''<sup>2</sup> − 1/4 ''y''<sup>2</sup>}}
|-
| [[File:Ellipse in coordinate system with semi-axes labelled.svg|150px]] <br>Points such that {{math|1=''Q''(''x'', ''y'') = 1}} <br> ([[Ellipse]]).
| [[File:Hyperbola2 SVG.svg|100x100px]] <br> Points such that {{math|1=''Q''(''x'', ''y'') = 1}} <br> ([[Hyperbola]]).
|}
A symmetric {{math|''n''×''n''}}-matrix is called ''[[positive-definite matrix|positive-definite]]'' (respectively negative-definite; indefinite), if for all nonzero vectors <math>x \in \mathbb{R}^n</math> the associated [[quadratic form]] given by
<math display="block" id="quadratic_forms">Q(\mathbf{x}) = \mathbf{x}^\mathsf{T} A \mathbf{x}</math>
takes only positive values (respectively only negative values; both some negative and some positive values).<ref>{{Harvard citations |last1=Horn |last2=Johnson |year=1985 |nb=yes |loc=Chapter 7 }}</ref> If the quadratic form takes only non-negative (respectively only non-positive) values, the symmetric matrix is called positive-semidefinite (respectively negative-semidefinite); hence the matrix is indefinite precisely when it is neither positive-semidefinite nor negative-semidefinite.

A symmetric matrix is positive-definite if and only if all its eigenvalues are positive.<ref>{{Harvard citations |last1=Horn |last2=Johnson |year=1985 |nb=yes |loc=Theorem 7.2.1 }}</ref> The table at the right shows two possibilities for 2×2 matrices.

Allowing as input two different vectors instead yields the [[bilinear form]] associated to {{mvar|A}}:<ref>{{Harvard citations |last1=Horn |last2=Johnson |year=1985 |nb=yes |loc=Example 4.0.6, p. 169 }}</ref>
<math display="block">B_A(\mathbf{x}, \mathbf{y}) = \mathbf{x}^\mathsf{T} A \mathbf{y}.</math>