zero matrix
{{Short description|Matrix whose entries are all 0}}
In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero. It also serves as the additive identity of the additive group of matrices, and is denoted by the symbol or followed by subscripts corresponding to the dimension of the matrix as the context sees fit.{{citation|title=Linear Algebra|series=Undergraduate Texts in Mathematics|first=Serge|last=Lang|authorlink=Serge Lang|publisher=Springer|year=1987|isbn=9780387964126|page=25|url=https://books.google.com/books?id=0DUXym7QWfYC&pg=PA25|quotation=We have a zero matrix in which aij = 0 for all i, j. ... We shall write it O.}}{{Cite web|title=Intro to zero matrices (article) {{!}} Matrices|url=https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:properties-of-matrix-addition-and-scalar-multiplication/a/intro-to-zero-matrices|access-date=2020-08-13|website=Khan Academy|language=en}}{{Cite web|last=Weisstein|first=Eric W.|title=Zero Matrix|url=https://mathworld.wolfram.com/ZeroMatrix.html|access-date=2020-08-13|website=mathworld.wolfram.com|language=en}} Some examples of zero matrices are
:
0_{1,1} = \begin{bmatrix}
0 \end{bmatrix}
,\
0_{2,2} = \begin{bmatrix}
0 & 0 \\
0 & 0 \end{bmatrix}
,\
0_{2,3} = \begin{bmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \end{bmatrix}
.\
Properties
The set of matrices with entries in a ring K forms a ring . The zero matrix in is the matrix with all entries equal to , where is the additive identity in K.
:
0_{K_{m,n}} = \begin{bmatrix}
0_K & 0_K & \cdots & 0_K \\
0_K & 0_K & \cdots & 0_K \\
\vdots & \vdots & \ddots & \vdots \\
0_K & 0_K & \cdots & 0_K \end{bmatrix}_{m \times n}
The zero matrix is the additive identity in .{{citation|title=Modern Algebra|first=Seth|last=Warner|publisher=Courier Dover Publications|year=1990|isbn=9780486663418|page=291|url=https://books.google.com/books?id=dT2KAAAAQBAJ&pg=PA291|quotation=The neutral element for addition is called the zero matrix, for all of its entries are zero.}} That is, for all it satisfies the equation
:
There is exactly one zero matrix of any given dimension m×n (with entries from a given ring), so when the context is clear, one often refers to the zero matrix. In general, the zero element of a ring is unique, and is typically denoted by 0 without any subscript indicating the parent ring. Hence the examples above represent zero matrices over any ring.
The zero matrix also represents the linear transformation which sends all the vectors to the zero vector.{{citation|title=Linear Algebra: An Introduction|first1=Richard|last1=Bronson|first2=Gabriel B.|last2=Costa|publisher=Academic Press|year=2007|isbn=9780120887842|page=377|url=https://books.google.com/books?id=ZErjtA3mIvkC&pg=PA377|quotation=The zero matrix represents the zero transformation 0, having the property 0(v) = 0 for every vector v ∈ V.}} It is idempotent, meaning that when it is multiplied by itself, the result is itself.
The zero matrix is the only matrix whose rank is 0.
Occurrences
In ordinary least squares regression, if there is a perfect fit to the data, the annihilator matrix is the zero matrix.
See also
- Identity matrix, the multiplicative identity for matrices
- Matrix of ones, a matrix where all elements are one
- Nilpotent matrix
- Single-entry matrix, a matrix where all but one element is zero