Inmathematics, particularlylinear algebra, azero matrix ornull matrix is amatrix all of whose entries arezero. It also serves as theadditive identity of theadditive 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.[1][2][3] Some examples of zero matrices are
The set of matrices with entries in aring K forms a ring. The zero matrix in is the matrix with all entries equal to, where is theadditive identity in K.
The zero matrix is the additive identity in.[4] That is, for all it satisfies the equation
There is exactly one zero matrix of any given dimensionm×n (with entries from a given ring), so when the context is clear, one often refers tothe zero matrix. In general, thezero element of a ring is unique, and is typically denoted by 0 without anysubscript indicating the parent ring. Hence the examples above represent zero matrices over any ring.
The zero matrix also represents thelinear transformation which sends all thevectors to thezero vector.[5] It isidempotent, meaning that when it is multiplied by itself, the result is itself.
The zero matrix is the only matrix whoserank is 0.
Inordinary least squares regression, if there is a perfect fit to the data, theannihilator matrix is the zero matrix.
We have a zero matrix in whichaij = 0 for alli, j. ... We shall write it O.
The neutral element for addition is called the zero matrix, for all of its entries are zero.
The zero matrix represents the zero transformation0, having the property0(v) = 0 for every vectorv ∈ V.
