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Difference between revisions of "Identity matrix"

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(expand somewhat)
(also unit matrix, cite Aitken (1939))
 
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''unit matrix''
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A square matrix $I$ with entries $1$ on the main diagonal and $0$ otherwise:
 
A square matrix $I$ with entries $1$ on the main diagonal and $0$ otherwise:
 
$$
 
$$
I_{ij} = \delta_{ij}
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I_{ij} = \delta_{ij} = \begin{cases} 1  & \text{if}\ i =j \\ 0 & \text{otherwise} \end{cases}  
 
$$
 
$$
 
where $\delta$ is the [[Kronecker symbol]].
 
where $\delta$ is the [[Kronecker symbol]].
  
 
If $R$ is a [[ring with identity]] and 0 and 1 are interpreted as elements of $R$, then $I$ is the [[identity element]] in the [[matrix ring]] $M_n(R)$.
 
If $R$ is a [[ring with identity]] and 0 and 1 are interpreted as elements of $R$, then $I$ is the [[identity element]] in the [[matrix ring]] $M_n(R)$.
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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}}

Latest revision as of 11:25, 2 April 2018

unit matrix

A square matrix $I$ with entries $1$ on the main diagonal and $0$ otherwise: $$ I_{ij} = \delta_{ij} = \begin{cases} 1 & \text{if}\ i =j \\ 0 & \text{otherwise} \end{cases} $$ where $\delta$ is the Kronecker symbol.

If $R$ is a ring with identity and 0 and 1 are interpreted as elements of $R$, then $I$ is the identity element in the matrix ring $M_n(R)$.


References

How to Cite This Entry:
Identity matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Identity_matrix&oldid=39112