Matrix unit: Difference between revisions
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{{Distinguish|unit matrix|unitary matrix|invertible matrix}} |
{{Distinguish|unit matrix|unitary matrix|invertible matrix}} |
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In [[linear algebra]], a '''matrix unit''' is a [[matrix (mathematics)|matrix]] with only one nonzero entry with value 1.<ref>{{cite book |last=Artin|first=Michael |title=Algebra |publisher= Prentice Hall|page=9}}</ref><ref name="pm">{{cite book |chapter=Chapter 17: Matrix Rings |title=Lectures on Modules and Rings |first=Tsit-Yuen |last=Lam |authorlink=Tsit-Yuen Lam |series=[[Graduate Texts in Mathematics]] |volume=189 |publisher=[[Springer Science+Business Media]] |year=1999 |pages=461–479}}</ref> The matrix unit with a 1 in the ''i''th row and ''j''th column is denoted as <math>E_{ij}</math>. For example, the 3 by 3 matrix unit with ''i'' = 1 and ''j'' = 2 is |
In [[linear algebra]], a '''matrix unit''' is a [[matrix (mathematics)|matrix]] with only one nonzero entry with value 1.<ref>{{cite book |last=Artin|first=Michael |title=Algebra |publisher= Prentice Hall|page=9}}</ref><ref name="pm">{{cite book |chapter=Chapter 17: Matrix Rings |title=Lectures on Modules and Rings |first=Tsit-Yuen |last=Lam |authorlink=Tsit-Yuen Lam |series=[[Graduate Texts in Mathematics]] |volume=189 |publisher=[[Springer Science+Business Media]] |year=1999 |pages=461–479}}</ref> The matrix unit with a 1 in the ''i''th row and ''j''th column is denoted as <math>E_{ij}</math>. For example, the 3 by 3 matrix unit with ''i'' = 1 and ''j'' = 2 is |
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<math display=block>E_{12} = \begin{bmatrix}0 & 1 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}</math> |
<math display=block>E_{12} = \begin{bmatrix}0 & 1 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}</math>A '''vector unit''' is a [[standard unit vector]]. |
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A '''single-entry matrix''' generalizes the matrix unit for matrices with only one nonzero entry of any value, not necessarily of value 1. |
A '''single-entry matrix''' generalizes the matrix unit for matrices with only one nonzero entry of any value, not necessarily of value 1. |
Revision as of 04:06, 17 July 2022
In linear algebra, a matrix unit is a matrix with only one nonzero entry with value 1.[1][2] The matrix unit with a 1 in the ith row and jth column is denoted as . For example, the 3 by 3 matrix unit with i = 1 and j = 2 is 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
The set of m by n matrix units is a basis of the space of m by n matrices.[2]
The product of two matrix units of the same square shape satisfies the relation where is the Kronecker delta.[2]
The group of scalar n-by-n matrices over a ring R is the centralizer of the subset of n-by-n matrix units in the set of n-by-n matrices over R.[2]
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]
References
- ^ Artin, Michael. Algebra. Prentice Hall. p. 9.
- ^ a b c d Lam, Tsit-Yuen (1999). "Chapter 17: Matrix Rings". Lectures on Modules and Rings. Graduate Texts in Mathematics. Vol. 189. Springer Science+Business Media. pp. 461–479.
- ^ Marcel Blattner (2009). "B-Rank: A top N Recommendation Algorithm". arXiv:0908.2741 [physics.data-an].