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Search
A p p e a r a n c e
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P a g e s f o r l o g g e d o u t e d i t o r s l e a r n m o r e ● C o n t r i b u t i o n s ● T a l k
( T o p )
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M a t r i x u n i t
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P r i n t / e x p o r t
● D o w n l o a d a s P D F ● P r i n t a b l e v e r s i o n
A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
In linear algebra , a matrix unit is a matrix with only one nonzero entry with value 1.[1] [2] i j
E
i j
{\displaystyle E_{ij}}
i j
E
12
=
[
0
1
0
0
0
0
0
0
0
]
{\displaystyle E_{12}={\begin{bmatrix}0&1&0\\0&0&0\\0&0&0\end{bmatrix}}}
A vector unit is a standard unit vector .
A single-entry matrix generalizes the matrix unit for matrices with only one nonzero entry of any value, not necessarily of value 1.
Properties [ edit ]
The set of m by n basis of the space of m by n [2]
The product of two matrix units of the same square shape
n ×
n
{\displaystyle n\times n}
E
i j
E
k l
=
δ
j k
E
i l
,
{\displaystyle E_{ij}E_{kl}=\delta _{jk}E_{il},}
δ
j k
{\displaystyle \delta _{jk}}
Kronecker delta .[2]
The group of scalar n n R centralizer of the subset of n n n n R [2]
The matrix norm (induced by the same two vector norms) of a matrix unit is equal to 1.
When multiplied by another matrix, it isolates a specific row or column in arbitrary position. For example, for any 3-by-3 matrix A [3]
E
23
A =
[
0
0
0
a
31
a
32
a
33
0
0
0
]
.
{\displaystyle E_{23}A=\left[{\begin{matrix}0&0&0\\a_{31}&a_{32}&a_{33}\\0&0&0\end{matrix}}\right].}
A
E
23
=
[
0
0
a
12
0
0
a
22
0
0
a
32
]
.
{\displaystyle AE_{23}=\left[{\begin{matrix}0&0&a_{12}\\0&0&a_{22}\\0&0&a_{32}\end{matrix}}\right].}
References [ edit ]
^ Marcel Blattner (2009). "B-Rank: A top N Recommendation Algorithm". arXiv :0908.2741 physics.data-an ].
t
e
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Matrix_unit&oldid=1133993681 " C a t e g o r i e s : ● S p a r s e m a t r i c e s ● 1 ( n u m b e r ) ● L i n e a r a l g e b r a s t u b s H i d d e n c a t e g o r i e s : ● A r t i c l e s w i t h s h o r t d e s c r i p t i o n ● S h o r t d e s c r i p t i o n i s d i f f e r e n t f r o m W i k i d a t a ● A l l s t u b a r t i c l e s
● T h i s p a g e w a s l a s t e d i t e d o n 1 6 J a n u a r y 2 0 2 3 , a t 1 3 : 4 4 ( U T C ) . ● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n . ● P r i v a c y p o l i c y ● A b o u t W i k i p e d i a ● D i s c l a i m e r s ● C o n t a c t W i k i p e d i a ● C o d e o f C o n d u c t ● D e v e l o p e r s ● S t a t i s t i c s ● C o o k i e s t a t e m e n t ● M o b i l e v i e w
Avector unit is a standard unit vector.
where 
is the Kronecker delta.[2]
![{\displaystyle E_{23}A=\left[{\begin{matrix}0&0&0\\a_{31}&a_{32}&a_{33}\\0&0&0\end{matrix}}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/465cc21ae93f6bf1cb956685cd85c81850c98760)
![{\displaystyle AE_{23}=\left[{\begin{matrix}0&0&a_{12}\\0&0&a_{22}\\0&0&a_{32}\end{matrix}}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a798c3f3f0e778a9847acfa5cc16eea6e1e69f7c)
