Matrix unit (original) (raw)

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Concept in mathematics

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 _i_th row and _j_th column is denoted as E i j {\displaystyle E_{ij}} {\displaystyle E_{ij}}. For example, the 3 by 3 matrix unit with i = 1 and j = 2 is 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}}} {\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.

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 n × n {\displaystyle n\times n} {\displaystyle n\times n} satisfies the relation E i j E k l = δ j k E i l , {\displaystyle E_{ij}E_{kl}=\delta _{jk}E_{il},} {\displaystyle E_{ij}E_{kl}=\delta _{jk}E_{il},}where δ j k {\displaystyle \delta _{jk}} {\displaystyle \delta _{jk}} 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]

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].} {\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].} {\displaystyle AE_{23}=\left[{\begin{matrix}0&0&a_{12}\\0&0&a_{22}\\0&0&a_{32}\end{matrix}}\right].}

  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. Vol. 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].