Editing Square matrix (section)

== Special kinds ==
{| class="wikitable" style="float:right; margin:0ex 0ex 2ex 2ex;"
|-
! Name !! Example with ''n'' = 3
|-
| [[Diagonal matrix]] || style="text-align:center;" | <math>
      \begin{bmatrix}
           a_{11} & 0      & 0 \\
           0      & a_{22} & 0 \\
           0      & 0      & a_{33}
      \end{bmatrix}
  </math>
|-
| [[Lower triangular matrix]] || style="text-align:center;" | <math>
      \begin{bmatrix}
           a_{11} & 0      & 0 \\
           a_{21} & a_{22} & 0 \\
           a_{31} & a_{32} & a_{33}
      \end{bmatrix}
  </math>
|-
| [[Upper triangular matrix]] || style="text-align:center;" | <math>
      \begin{bmatrix}
           a_{11} & a_{12} & a_{13} \\
           0      & a_{22} & a_{23} \\
           0      & 0      & a_{33}
      \end{bmatrix}
  </math>
|}
=== Diagonal or triangular matrix ===
If all entries outside the main diagonal are zero, <math>A</math> is called a [[diagonal matrix]]. If all entries below (resp. above) the main diagonal are zero, <math>A</math> is called an upper (resp. lower) [[triangular matrix]].

=== Identity matrix ===
The [[identity matrix]] <math>I_n</math> of size <math>n</math> is the <math>n \times n</math> matrix in which all the elements on the [[main diagonal]] are equal to 1 and all other elements are equal to 0, e.g.
<math display="block">
I_1 = \begin{bmatrix} 1 \end{bmatrix}
,\ 
I_2 = \begin{bmatrix}
         1 & 0 \\
         0 & 1 
      \end{bmatrix}
,\ \ldots ,\ 
I_n = \begin{bmatrix}
         1 & 0 & \cdots & 0 \\
         0 & 1 & \cdots & 0 \\
         \vdots & \vdots & \ddots & \vdots \\
         0 & 0 & \cdots & 1
      \end{bmatrix}.
</math>
It is a square matrix of order {{nowrap|<math>n</math>,}} and also a special kind of [[diagonal matrix]]. The term ''identity matrix'' refers to the property of matrix multiplication that
<math display=block>I_m A = A I_n = A</math> for any <math>m \times n</math> matrix {{nowrap|<math>A</math>.}}

===Invertible matrix and its inverse===
A square matrix <math>A</math> is called ''[[invertible matrix|invertible]]'' or ''non-singular'' if there exists a matrix <math>B</math> such that<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Definition I.2.28 }}</ref><ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Definition I.5.13 }}</ref>
<math display="block">AB = BA = I_n.</math>
If <math>B</math> exists, it is unique and is called the ''[[inverse matrix]]'' of {{nowrap|<math>A</math>,}} denoted {{nowrap|<math>A^{-1}</math>.}}

===Symmetric or skew-symmetric matrix===
A square matrix <math>A</math> that is equal to its transpose, i.e., {{nowrap|<math>A^{\mathsf T} = A</math>,}} is a [[symmetric matrix]].  If instead {{nowrap|<math>A^{\mathsf T} = -A</math>,}} then <math>A</math> is called a [[skew-symmetric matrix]]. 

For a complex square matrix {{nowrap|<math>A</math>,}} often the appropriate analogue of the transpose is the [[conjugate transpose]] {{nowrap|<math>A^*</math>,}} defined as the transpose of the [[complex conjugate]] of {{nowrap|<math>A</math>.}}  A complex square matrix <math>A</math> satisfying <math>A^*=A</math> is called a [[Hermitian matrix]].  If instead {{nowrap|<math>A^* = -A</math>,}} then <math>A</math> is called a [[skew-Hermitian matrix]].

By the [[spectral theorem]], real symmetric (or complex Hermitian) matrices have an orthogonal (or unitary) [[eigenbasis]]; i.e., every vector is expressible as a [[linear combination]] of eigenvectors. In both cases, all eigenvalues are real.<ref>{{Harvard citations |last1=Horn |last2=Johnson |year=1985 |nb=yes |loc=Theorem 2.5.6 }}</ref>

===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>

=== Orthogonal matrix ===
An ''[[orthogonal matrix]]'' is a [[Matrix (mathematics)#Square matrices|square matrix]] with [[real number|real]] entries whose columns and rows are [[orthogonal]] [[unit vector]]s (i.e., [[orthonormality|orthonormal]] vectors). Equivalently, a matrix ''A'' is orthogonal if its [[transpose]] is equal to its [[inverse matrix|inverse]]:
<math display="block">A^\textsf{T} = A^{-1}, </math>
which entails
<math display="block">A^\textsf{T} A = A A^\textsf{T} = I, </math>
where ''I'' is the [[identity matrix]]. 

An orthogonal matrix {{mvar|A}} is necessarily [[Invertible matrix|invertible]] (with inverse {{math|1=''A''<sup>−1</sup> = ''A''<sup>T</sup>}}), [[Unitary matrix|unitary]] ({{math|1=''A''<sup>−1</sup> = ''A''*}}), and [[Normal matrix|normal]] ({{math|1=''A''*''A'' = ''AA''*}}). The [[determinant]] of any orthogonal matrix is either +1 or −1.  The [[special orthogonal group]] <math>\operatorname{SO}(n)</math> consists of the {{math|''n'' × ''n''}} orthogonal matrices with [[determinant]] +1. 

The [[complex number|complex]] analogue of an orthogonal matrix is a [[unitary matrix]].

===Normal matrix===
A real or complex square matrix <math>A</math> is called ''[[normal matrix|normal]]'' if {{nowrap|<math>A^* A = AA^*</math>.}}  If a real square matrix is symmetric, skew-symmetric, or orthogonal, then it is normal. If a complex square matrix is Hermitian, skew-Hermitian, or unitary, then it is normal. Normal matrices are of interest mainly because they include the types of matrices just listed and form the broadest class of matrices for which the [[spectral theorem]] holds.<ref>Artin, ''Algebra'', 2nd edition, Pearson, 2018, section 8.6.</ref>