Matrix unit

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 $E_{ij}$ . For example, the 3 by 3 matrix unit with i = 1 and j = 2 is $E_{12}={\begin{bmatrix}0&1&0\\0&0&0\\0&0&0\end{bmatrix}}$ 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.

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 $n\times n$ satisfies the relation $E_{ij}E_{kl}=\delta _{jk}E_{il},$ where $\delta _{jk}$ 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=\left[{\begin{matrix}0&0&0\\a_{31}&a_{32}&a_{33}\\0&0&0\end{matrix}}\right].

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

References

^ 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].

v t e Matrix classes
Explicitly constrained entries	Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal 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 Matrix unit Metzler Moore Nonnegative Pentadiagonal Permutation Persymmetric Polynomial Quaternionic Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Vandermonde Walsh Z
Constant	Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero
Conditions on eigenvalues or eigenvectors	Companion Convergent Defective Definite Diagonalizable Hurwitz-stable Positive-definite Stieltjes
Satisfying conditions on products or inverses	Congruent Idempotent or Projection Invertible Involutory Nilpotent Normal Orthogonal Unimodular Unipotent Unitary Totally unimodular Weighing
With specific applications	Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Confusion Coxeter Distance Duplication and elimination Euclidean distance Fundamental (linear differential equation) Generator Gram Hessian Householder Jacobian Moment Payoff Pick Random Rotation Routh-Hurwitz Seifert Shear Similarity Symplectic Totally positive Transformation
Used in statistics	Centering Correlation Covariance Design Doubly stochastic Fisher information Hat Precision Stochastic Transition
Used in graph theory	Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel 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 normal form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Row echelon form Wronskian
Mathematics portal List of matrices Category:Matrices

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