Matrix unit

Short description: Concept in mathematics

Not to be confused with unit matrix, unitary matrix, or invertible matrix.

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 Eij. For example, the 3 by 3 matrix unit with i = 1 and j = 2 is E12=[010000000]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

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 satisfies the relation EijEkl=δjkEil, where δjk 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]

E23A=[000a31a32a33000].
AE23=[00a1200a2200a32].

## References

1. ↑ Artin, Michael. Algebra. Prentice Hall. p. 9.
2. ↑ 2.0 2.1 2.2 2.3 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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Matrix classes

Explicitly constrained entries|

* (0,1)
* Alternant
* Anti-diagonal
* Anti-Hermitian
* Anti-symmetric
* Arrowhead
* Band
* Bidiagonal
* Binary
* Bisymmetric
* Block-diagonal
* Block
* Block tridiagonal
* Boolean
* Cauchy
* Centrosymmetric
* Conference
* Complex Hadamard
* Copositive
* Diagonally dominant
* Diagonal
* Discrete Fourier Transform
* Elementary
* Equivalent
* Frobenius
* Generalized permutation
* Hadamard
* Hankel
* Hermitian
* Hessenberg
* Hollow
* Integer
* Logical
* Markov
* Metzler
* Monomial
* Moore
* Nonnegative
* Partitioned
* Parisi
* Pentadiagonal
* Permutation
* Persymmetric
* Polynomial
* Positive
* Quaternionic
* Sign
* Signature
* Skew-Hermitian
* Skew-symmetric
* Skyline
* Sparse
* Sylvester
* Symmetric
* Toeplitz
* Triangular
* Tridiagonal
* Unitary
* Vandermonde
* Walsh
* Z

Constant|

* Exchange
* Hilbert
* Identity
* Lehmer
* Of ones
* Pascal
* Pauli
* Redheffer
* Shift
* Zero

Conditions on eigenvalues or eigenvectors|

* Companion
* Convergent
* Defective
* Diagonalizable
* Hurwitz
* Positive-definite
* Stability
* Stieltjes

Satisfying conditions on products or inverses|

* Congruent
* Idempotent or Projection
* Invertible
* Involutory
* Nilpotent
* Normal
* Orthogonal
* Orthonormal
* Singular
* Unimodular
* Unipotent
* Totally unimodular
* Weighing

With specific applications|

* Adjugate
* Alternating sign
* Augmented
* Bézout
* Carleman
* Cartan
* Circulant
* Cofactor
* Commutation
* Confusion
* Coxeter
* Derogatory
* Distance
* Duplication
* Elimination
* Euclidean distance
* Fundamental (linear differential equation)
* Generator
* Gramian
* Hessian
* Householder
* Jacobian
* Moment
* Payoff
* Pick
* Random
* Rotation
* Seifert
* Shear
* Similarity
* Symplectic
* Totally positive
* Transformation
* Wedderburn
* X–Y–Z

Used in statistics|

* Bernoulli
* Centering
* Correlation
* Covariance
* Design
* Dispersion
* Doubly stochastic
* Fisher information
* Hat
* Precision
* Stochastic
* Transition

Used in graph theory|

* Adjacency
* Biadjacency
* Degree
* Edmonds
* Incidence
* Laplacian
* Seidel adjacency
* Skew-adjacency
* Tutte

Used in science and engineering|

* Cabibbo–Kobayashi–Maskawa
* Density
* Fundamental (computer vision)
* Fuzzy associative
* Gamma
* Gell-Mann
* Hamiltonian
* Irregular
* Overlap
* S
* State transition
* Substitution
* Z (chemistry)

Related terms|

* Jordan canonical form
* Linear independence
* Matrix exponential
* Matrix representation of conic sections
* Perfect matrix
* Pseudoinverse
* Quaternionic matrix
* Row echelon form
* Wronskian

* List of matrices
* Category:Matrices

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