Zero matrix

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.[1][2][3][4] Some examples of zero matrices are

PropertiesEdit

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.

 

The zero matrix is the additive identity in  .[5] 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.[6] 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.

OccurrencesEdit

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.[7]

In ordinary least squares regression, if there is a perfect fit to the data, the annihilator matrix is the zero matrix.

See alsoEdit

ReferencesEdit

  1. ^ "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-13.
  2. ^ Lang, Serge (1987), Linear Algebra, Undergraduate Texts in Mathematics, Springer, p. 25, ISBN 9780387964126, We have a zero matrix in which aij = 0 for all ij. ... We shall write it O. CS1 maint: discouraged parameter (link)
  3. ^ "Intro to zero matrices (article) | Matrices". Khan Academy. Retrieved 2020-08-13.
  4. ^ Weisstein, Eric W. "Zero Matrix". mathworld.wolfram.com. Retrieved 2020-08-13.
  5. ^ Warner, Seth (1990), Modern Algebra, Courier Dover Publications, p. 291, ISBN 9780486663418, The neutral element for addition is called the zero matrix, for all of its entries are zero.
  6. ^ Bronson, Richard; Costa, Gabriel B. (2007), Linear Algebra: An Introduction, Academic Press, p. 377, ISBN 9780120887842, The zero matrix represents the zero transformation 0, having the property 0(v) = 0 for every vector v ∈ V.
  7. ^ Cassaigne, Julien; Halava, Vesa; Harju, Tero; Nicolas, Francois (2014). "Tighter Undecidability Bounds for Matrix Mortality, Zero-in-the-Corner Problems, and More". arXiv:1404.0644 [cs.DM].