Matrix unit

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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

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[]

  1. ^ Artin, Michael. Algebra. Prentice Hall. p. 9.
  2. ^ a b c d Lam, Tsit-Yuen (1999). "Chapter 17: Matrix Rings". Lectures on Modules and Rings. Graduate Texts in Mathematics. 189. Springer Science+Business Media. pp. 461–479.
  3. ^ Marcel Blattner (2009). "B-Rank: A top N Recommendation Algorithm". arXiv:0908.2741 [physics.data-an].
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