Matrix unit: Difference between revisions
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{{Short description|Concept in mathematics}} |
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{{unreferenced|date=October 2007}} |
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{{Distinguish|unit matrix|unitary matrix|invertible matrix}} |
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In [[mathematics]], a '''matrix unit''' is an idealisation of the concept of a [[Matrix (mathematics)|matrix]], with a focus on the algebraic properties of [[matrix multiplication]]. The topic is comparatively obscure within [[linear algebra]], because it entirely ignores the numeric properties of matrices; it is mostly encountered in the context of [[abstract algebra]], especially the theory of [[semigroup]]s. |
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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 |
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<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. |
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Despite the name, matrix units are not the same as [[Identity matrix|unit matrices]] or [[Unitary matrix|unitary matrices]]. |
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== Properties == |
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Two matrices can be multiplied when the number of columns in one is the same as the number of rows in the other; otherwise, they are incompatible. The idea behind matrix units is to look at this fact in isolation: a matrix unit is a matrix with dimensions, but with the entries scooped out. |
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The set of ''m'' by ''n'' matrix units is a [[basis (linear algebra)|basis]] of the space of ''m'' by ''n'' matrices.<ref name="pm"/> |
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Let ''I'' be a nonempty [[Set (mathematics)|set]], to be used for counting the matrix rows and columns. There is no requirement for it to be finite; indeed, standard matrix algebra would use the set of [[natural number]]s (not including zero) '''N'''<sup>+</sup>. A matrix unit is either an [[ordered pair]] (''r'', ''c''), with ''r'' and ''c'' elements of ''I'', or it is a special "zero" object, written as "0". Multiplication is defined as follows: |
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* 0 ''x'' = ''x'' 0 = 0 for any matrix unit ''x''; |
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* (''r'', ''c'') (''s'', ''d'') = (''r'', ''d'') if ''c'' = ''s'', and 0 if ''c'' ≠ ''s''. |
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The 0 element can be seen as an "error symbol" for when multiplication fails; the first rule implies that errors propagate through an entire product containing a single incompatible combination. |
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The product of two matrix units of the same square shape <math>n \times n</math> satisfies the relation |
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For example, the product (with ''I'' = '''N'''<sup>+</sup>) |
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<math display=block>E_{ij}E_{kl} = \delta_{jk}E_{il},</math> |
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:(2, 3) (3, 2) (2, 1) (1, 4) = (2, 4) |
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where <math>\delta_{jk}</math> is the [[Kronecker delta]].<ref name="pm"/> |
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represents the abstract matrix multiplication |
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\begin{bmatrix} |
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\cdot & \cdot & \cdot \\ |
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\cdot & \cdot & \cdot \\ |
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\end{bmatrix} |
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\begin{bmatrix} |
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\cdot & \cdot \\ |
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\cdot & \cdot \\ |
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\cdot & \cdot \\ |
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\end{bmatrix} |
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\begin{bmatrix} |
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\cdot \\ |
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\cdot \\ |
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\end{bmatrix} |
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\begin{bmatrix} |
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\cdot & \cdot & \cdot & \cdot \\ |
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\end{bmatrix} |
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= |
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\begin{bmatrix} |
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\cdot & \cdot & \cdot & \cdot \\ |
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\cdot & \cdot & \cdot & \cdot \\ |
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\end{bmatrix} |
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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"/> |
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Another notation for (''r'', ''c'') is ''A''<sub>''r c''</sub>, following the convention for naming a single entry of a matrix. (Different letters are used in the "''A''" position to refer to matrix units on a different base set.) The composition rule may be expressed using the [[Kronecker delta]] as |
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: ''X''<sub>''r c''</sub> ''X''<sub>''s d''</sub> = δ<sub>''c s''</sub> ''X''<sub>''r d''</sub>. |
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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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With these rules, (''I'' × ''I'') ∪ {0} is a semigroup with zero. Its construction is analogous to that for other important semigroups, such as [[rectangular band]]s and [[Rees matrix semigroup]]s. It also arises as the [[Trace (semigroup theory)|trace]] of the unique [[Green's relations|''D''-class]] of the [[bicyclic semigroup]], meaning that it summarises how composition for members of that class interacts with the structure of the semigroup's [[principal ideal]]s. |
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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 |
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A semigroup of matrix units is [[0-simple]], because any two nonzero elements generate the same two-sided ideal (the entire semigroup), and the semigroup is non-null. The elements (''r'', ''c'') and (''s'', ''d'') are ''D''-related via |
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| author = Marcel Blattner |
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: (''r'', ''c'') ''R'' (''r'', ''d'') ''L'' (''s'', ''d''), |
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| title = B-Rank: A top N Recommendation Algorithm |
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as any pairs are ''R''-related if they have the same first coordinate and ''L''-related if they have the same second coordinate. All ''H''-classes are singletons. The [[idempotent]]s are the "square" matrix units (''a'', ''a'') for ''a'' in ''I'', together with 0. |
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| year = 2009 |
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| class = physics.data-an |
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| eprint = 0908.2741 |
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}}</ref> |
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E_{23}A = \left[ \begin{matrix} 0 & 0& 0 \\ a_{31} & a_{32} & a_{33} \\ 0 & 0 & 0 \end{matrix}\right]. |
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: <math> |
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AE_{23} = \left[ \begin{matrix} 0 & 0 & a_{12} \\ 0 & 0 & a_{22} \\ 0 & 0 & a_{32} \end{matrix}\right]. |
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</math> |
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==References== |
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{{reflist}} |
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{{matrix classes}} |
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{{Linear-algebra-stub}} |
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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 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 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]
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].
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