Namespaces
Variants
Actions

Difference between revisions of "Hermitian matrix"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
''Hermitian-symmetric matrix, self-conjugate matrix''
+
<!--
 +
h0470701.png
 +
$#A+1 = 20 n = 0
 +
$#C+1 = 20 : ~/encyclopedia/old_files/data/H047/H.0407070 Hermitian matrix,
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
A square matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047070/h0470701.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047070/h0470702.png" /> that is the same as its Hermitian-conjugate matrix
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/h/h047/h047070/h0470703.png" /></td> </tr></table>
+
''Hermitian-symmetric matrix, self-conjugate matrix''
  
that is, a matrix whose entries satisfy the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047070/h0470704.png" />. If all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047070/h0470705.png" />, then a Hermitian matrix is symmetric (cf. [[Symmetric matrix|Symmetric matrix]]). The Hermitian matrices of a fixed order form a vector space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047070/h0470706.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047070/h0470707.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047070/h0470708.png" /> are two Hermitian matrices of the same order, then so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047070/h0470709.png" />. Under the operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047070/h04707010.png" /> the Hermitian matrices (of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047070/h04707011.png" />) form a [[Jordan algebra|Jordan algebra]]. The product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047070/h04707012.png" /> of two Hermitian matrices is itself Hermitian if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047070/h04707013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047070/h04707014.png" /> commute.
+
A square matrix $  A = \| a _ {ik} \| $
 +
over $  \mathbf C $
 +
that is the same as its Hermitian-conjugate matrix
  
The Hermitian matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047070/h04707015.png" /> are the matrices of Hermitian transformations of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047070/h04707016.png" />-dimensional unitary space in an orthonormal basis (see [[Self-adjoint linear transformation|Self-adjoint linear transformation]]). On the other hand, Hermitian matrices are the matrices of Hermitian forms in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047070/h04707017.png" />-dimensional complex vector space. Like Hermitian forms (cf. [[Hermitian form|Hermitian form]]), Hermitian matrices can be defined over any skew-field with an anti-involution.
+
$$
 +
A  ^ {*}  = \overline{A}\; {}  ^ {T}  = \| \overline{ {a _ {ki} }}\; \| ,
 +
$$
  
All eigen values of a Hermitian matrix are real. For every Hermitian matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047070/h04707018.png" /> there exists a unitary matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047070/h04707019.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047070/h04707020.png" /> is a real diagonal matrix. A Hermitian matrix is called non-negative (or positive semi-definite) if all its principal minors are non-negative, and positive definite if they are all positive. Non-negative (positive-definite) Hermitian matrices correspond to non-negative (positive-definite) Hermitian linear transformations and Hermitian forms.
+
that is, a matrix whose entries satisfy the condition  $  a _ {ik} = \overline{ {a _ {ki} }}\; $.  
 +
If all the  $  a _ {ik} \in \mathbf R $,
 +
then a Hermitian matrix is symmetric (cf. [[Symmetric matrix|Symmetric matrix]]). The Hermitian matrices of a fixed order form a vector space over  $  \mathbf R $.  
 +
If  $  A $
 +
and  $  B $
 +
are two Hermitian matrices of the same order, then so is $  AB + BA $.  
 +
Under the operation  $  A \cdot B = ( AB + BA ) / 2 $
 +
the Hermitian matrices (of order  $  n $)  
 +
form a [[Jordan algebra|Jordan algebra]]. The product  $  AB $
 +
of two Hermitian matrices is itself Hermitian if and only if  $  A $
 +
and $  B $
 +
commute.
  
 +
The Hermitian matrices of order  $  n $
 +
are the matrices of Hermitian transformations of an  $  n $-
 +
dimensional unitary space in an orthonormal basis (see [[Self-adjoint linear transformation|Self-adjoint linear transformation]]). On the other hand, Hermitian matrices are the matrices of Hermitian forms in an  $  n $-
 +
dimensional complex vector space. Like Hermitian forms (cf. [[Hermitian form|Hermitian form]]), Hermitian matrices can be defined over any skew-field with an anti-involution.
  
 +
All eigen values of a Hermitian matrix are real. For every Hermitian matrix  $  A $
 +
there exists a unitary matrix  $  U $
 +
such that  $  U  ^ {-} 1 AU $
 +
is a real diagonal matrix. A Hermitian matrix is called non-negative (or positive semi-definite) if all its principal minors are non-negative, and positive definite if they are all positive. Non-negative (positive-definite) Hermitian matrices correspond to non-negative (positive-definite) Hermitian linear transformations and Hermitian forms.
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,  "Matrix theory" , '''1–2''' , Chelsea, reprint  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Noble,  J.W. Daniel,  "Applied linear algebra" , Prentice-Hall  (1979)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,  "Matrix theory" , '''1–2''' , Chelsea, reprint  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Noble,  J.W. Daniel,  "Applied linear algebra" , Prentice-Hall  (1979)</TD></TR></table>

Latest revision as of 22:10, 5 June 2020


Hermitian-symmetric matrix, self-conjugate matrix

A square matrix $ A = \| a _ {ik} \| $ over $ \mathbf C $ that is the same as its Hermitian-conjugate matrix

$$ A ^ {*} = \overline{A}\; {} ^ {T} = \| \overline{ {a _ {ki} }}\; \| , $$

that is, a matrix whose entries satisfy the condition $ a _ {ik} = \overline{ {a _ {ki} }}\; $. If all the $ a _ {ik} \in \mathbf R $, then a Hermitian matrix is symmetric (cf. Symmetric matrix). The Hermitian matrices of a fixed order form a vector space over $ \mathbf R $. If $ A $ and $ B $ are two Hermitian matrices of the same order, then so is $ AB + BA $. Under the operation $ A \cdot B = ( AB + BA ) / 2 $ the Hermitian matrices (of order $ n $) form a Jordan algebra. The product $ AB $ of two Hermitian matrices is itself Hermitian if and only if $ A $ and $ B $ commute.

The Hermitian matrices of order $ n $ are the matrices of Hermitian transformations of an $ n $- dimensional unitary space in an orthonormal basis (see Self-adjoint linear transformation). On the other hand, Hermitian matrices are the matrices of Hermitian forms in an $ n $- dimensional complex vector space. Like Hermitian forms (cf. Hermitian form), Hermitian matrices can be defined over any skew-field with an anti-involution.

All eigen values of a Hermitian matrix are real. For every Hermitian matrix $ A $ there exists a unitary matrix $ U $ such that $ U ^ {-} 1 AU $ is a real diagonal matrix. A Hermitian matrix is called non-negative (or positive semi-definite) if all its principal minors are non-negative, and positive definite if they are all positive. Non-negative (positive-definite) Hermitian matrices correspond to non-negative (positive-definite) Hermitian linear transformations and Hermitian forms.

Comments

References

[a1] F.R. [F.R. Gantmakher] Gantmacher, "Matrix theory" , 1–2 , Chelsea, reprint (1959) (Translated from Russian)
[a2] B. Noble, J.W. Daniel, "Applied linear algebra" , Prentice-Hall (1979)
How to Cite This Entry:
Hermitian matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermitian_matrix&oldid=14435
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article