Zero matrix

In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero.[1] Some examples of zero matrices are

Properties

The set of m×n 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 .[2] That is, for all it satisfies

There is exactly one zero matrix of any given size m×n having entries in 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 typically denoted as 0 without any subscript indicating the parent ring. Hence the examples above represent zero matrices over any ring.

The zero matrix represents the linear transformation sending all vectors to the zero vector.[3]

The zero matrix 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

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.[4]

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

See also

References

  1. ↑ 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 i, j. ... We shall write it O.
  2. ↑ 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.
  3. ↑ 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.
  4. ↑ 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 Freely accessible [cs.DM].

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