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Difference between revisions of "Diagonal matrix"

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''quasi-scalar matrix''
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A square matrix in which all entries — with the possible exception of the elements on the main diagonal — are zero.
 
A square matrix in which all entries — with the possible exception of the elements on the main diagonal — are zero.
  
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====Comments====
 
====Comments====
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I.e. an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031490/d0314901.png" /> diagonal matrix over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031490/d0314902.png" /> has the form
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I.e. an $(n\times n)$ diagonal matrix over a field $K$ has the form
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$$\begin{pmatrix}a_1&0&\cdots&0\\0&a_2&\cdots&0\\\vdots&\vdots&\ddots&\vdots\\0&\cdots&\cdots&a_n\end{pmatrix},$$
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where the $a_i$ are elements of $K$.
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See also: [[Defective matrix]].
  
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<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/d/d031/d031490/d0314903.png" /></td> </tr></table>
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====References====
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* A.C. Aitken, "Determinants and matrices", Oliver and Boyd (1939)  {{ZBL|65.1111.05}} {{ZBL|0022.10005}}
  
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where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031490/d0314904.png" /> are elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031490/d0314905.png" />.
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[[Category:Special matrices]]

Latest revision as of 03:37, 25 February 2022

2020 Mathematics Subject Classification: Primary: 15B [MSN][ZBL]

quasi-scalar matrix

A square matrix in which all entries — with the possible exception of the elements on the main diagonal — are zero.


Comments

I.e. an $(n\times n)$ diagonal matrix over a field $K$ has the form

$$\begin{pmatrix}a_1&0&\cdots&0\\0&a_2&\cdots&0\\\vdots&\vdots&\ddots&\vdots\\0&\cdots&\cdots&a_n\end{pmatrix},$$

where the $a_i$ are elements of $K$.

See also: Defective matrix.

References

How to Cite This Entry:
Diagonal matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diagonal_matrix&oldid=12142
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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