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 [math]\displaystyle{ m \times n }[/math] matrices, and is denoted by the symbol [math]\displaystyle{ O }[/math] or [math]\displaystyle{ 0 }[/math] followed by subscripts corresponding to the dimension of the matrix as the context sees fit.[1][2][3] Some examples of zero matrices are
- [math]\displaystyle{ 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} .\ }[/math]
Properties
The set of [math]\displaystyle{ m \times n }[/math] matrices with entries in a ring K forms a ring [math]\displaystyle{ K_{m,n} }[/math]. The zero matrix [math]\displaystyle{ 0_{K_{m,n}} \, }[/math] in [math]\displaystyle{ K_{m,n} \, }[/math] is the matrix with all entries equal to [math]\displaystyle{ 0_K \, }[/math], where [math]\displaystyle{ 0_K }[/math] is the additive identity in K.
- [math]\displaystyle{ 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} }[/math]
The zero matrix is the additive identity in [math]\displaystyle{ K_{m,n} \, }[/math].[4] That is, for all [math]\displaystyle{ A \in K_{m,n} \, }[/math] it satisfies the equation
- [math]\displaystyle{ 0_{K_{m,n}}+A = A + 0_{K_{m,n}} = A. }[/math]
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.[5] 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
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.[6]
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
References
- ↑ Lang, Serge (1987), Linear Algebra, Undergraduate Texts in Mathematics, Springer, p. 25, ISBN 9780387964126, https://books.google.com/books?id=0DUXym7QWfYC&pg=PA25
- ↑ "Intro to zero matrices (article) | Matrices" (in en). https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:properties-of-matrix-addition-and-scalar-multiplication/a/intro-to-zero-matrices.
- ↑ Weisstein, Eric W.. "Zero Matrix" (in en). https://mathworld.wolfram.com/ZeroMatrix.html.
- ↑ Warner, Seth (1990), Modern Algebra, Courier Dover Publications, p. 291, ISBN 9780486663418, https://books.google.com/books?id=dT2KAAAAQBAJ&pg=PA291
- ↑ Bronson, Richard; Costa, Gabriel B. (2007), Linear Algebra: An Introduction, Academic Press, p. 377, ISBN 9780120887842, https://books.google.com/books?id=ZErjtA3mIvkC&pg=PA377
- ↑ 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].
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