Matrix unit: Difference between revisions
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The group of [[scalar matrix|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''.<ref name="pm"/> |
The group of [[scalar matrix|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''.<ref name="pm"/> |
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The [[matrix norm]] (induced by the same two vector norms) of a matrix unit is equal to 1. |
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When multiplied by another matrix, it isolates a specific row or column in arbitrary position. For example, for any 3-by-3 matrix ''A'':<ref>{{Cite arXiv |
When multiplied by another matrix, it isolates a specific row or column in arbitrary position. For example, for any 3-by-3 matrix ''A'':<ref>{{Cite arXiv |
Latest revision as of 13:44, 16 January 2023
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 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 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
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]
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]
References[edit]
- ^ 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].