linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...

, a matrix unit is a

matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...

with only one nonzero entry with value 1. The matrix unit with a 1 in the ''i''th row and ''j''th column is denoted as

E_

. For example, the 3 by 3 matrix unit with ''i'' = 1 and ''j'' = 2 is

E_ = \begin0 & 1 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0 \end

A vector unit is a

standard unit vector In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors, each of whose components are all zero, except one that equals 1. For exampl ...

. 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 Basis is a term used in mathematics, finance, science, and other contexts to refer to foundational concepts, valuation measures, or organizational names; here, it may refer to: Finance and accounting * Adjusted basis, the net cost of an asse ...

of the space of ''m'' by ''n'' matrices. The product of two matrix units of the same square shape

n \times n

satisfies the relation

E_E_ = \delta_E_,

where

\delta_

is the

Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...

. The group of

scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...

''n''-by-''n'' matrices over a ring ''R'' is the

centralizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of ele ...

of the subset of ''n''-by-''n'' matrix units in the set of ''n''-by-''n'' matrices over ''R''. The

matrix norm In the field of mathematics, norms are defined for elements within a vector space. Specifically, when the vector space comprises matrices, such norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also ...

(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'': :

E_A = \left \begin 0 & 0& 0 \\ a_ & a_ & a_ \\ 0 & 0 & 0 \end\right

AE_ = \left \begin 0 & 0 & a_ \\ 0 & 0 & a_ \\ 0 & 0 & a_  \end\right

References

Sparse matrices 1 (number) {{Linear-algebra-stub