mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, particularly

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 matrices. ...

, a zero matrix or null matrix is a

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

all of whose entries are

zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...

. It also serves as the

additive identity In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element ''x'' in the set, yields ''x''. One of the most familiar additive identities is the number 0 from element ...

of the

additive group An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structure ...

m \times n

matrices, and is denoted by the symbol

O

0

followed by subscripts corresponding to the dimension of the matrix as the context sees fit. Some examples of zero matrices are :

0_ = \begin
0 \end
,\ 
0_ = \begin
0 & 0 \\
0 & 0 \end
,\ 
0_ = \begin
0 & 0 & 0 \\
0 & 0 & 0 \end
.\

Properties

The set of

m \times n

matrices with entries in a

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

K forms a ring

K_

. The zero matrix

0_ \,

K_ \,

is the matrix with all entries equal to

0_K \,

, where

0_K

is the

in K. :

0_ = \begin
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_

The zero matrix is the additive identity in

K_ \,

. That is, for all

A \in K_ \,

it satisfies the equation :

0_+A = A + 0_ = A.

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 In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An additive identi ...

of a ring is unique, and is typically denoted by 0 without any

subscript A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the baseline, whil ...

indicating the parent ring. Hence the examples above represent zero matrices over any ring. The zero matrix also represents the

linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...

which sends all the

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

s to the

zero vector In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An additive identi ...

. It is

idempotent Idempotence (, ) is the property of certain operation (mathematics), operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence ...

, meaning that when it is multiplied by itself, the result is itself. The zero matrix is the only matrix whose

rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...

is 0.

Occurrences

The mortal matrix problem is the problem of determining, given a finite set of ''n'' × ''n'' matrices with integer entries, whether they can be multiplied in some order, possibly with repetition, to yield the zero matrix. This is known to be undecidable for a set of six or more 3 × 3 matrices, or a set of two 15 × 15 matrices. In

ordinary least squares In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the prin ...

regression, if there is a perfect fit to the data, the

annihilator matrix In statistics, the projection matrix (\mathbf), sometimes also called the influence matrix or hat matrix (\mathbf), maps the vector of response variable, response values (dependent variable values) to the vector of fitted values (or predicted value ...

is the zero matrix.

References

{{Matrix classes Matrices 0 (number) Sparse matrices

Properties

Occurrences

See also

References