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Zero matrix
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<h2 id="properties">Properties</h2> <p>The set of m × n {\displaystyle m\times n} matrices with entries in a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> K forms a ring K m , n {\displaystyle K_{m,n}} . The zero matrix 0 K m , n {\displaystyle 0_{K_{m,n}}\,} in K m , n {\displaystyle K_{m,n}\,} is the matrix with all entries equal to 0 K {\displaystyle 0_{K}\,} , where 0 K {\displaystyle 0_{K}} is the <a href="/facts/Additive_identity/wNDDkXJe">additive identity</a> in K. </p> 0 K m , n = [ 0 K 0 K ⋯ 0 K 0 K 0 K ⋯ 0 K ⋮ ⋮ ⋱ ⋮ 0 K 0 K ⋯ 0 K ] m × n {\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}} <p>The zero matrix is the additive identity in K m , n {\displaystyle K_{m,n}\,} .<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> That is, for all A ∈ K m , n {\displaystyle A\in K_{m,n}\,} it satisfies the equation </p> 0 K m , n + A = A + 0 K m , n = A . {\displaystyle 0_{K_{m,n}}+A=A+0_{K_{m,n}}=A.} <p>There is exactly one zero matrix of any given dimension <i>m</i>×<i>n</i> (with entries from a given ring), so when the context is clear, one often refers to <i>the</i> zero matrix. In general, the <a href="/facts/Zero_element/PvH2qhxh">zero element</a> of a ring is unique, and is typically denoted by 0 without any <a href="/facts/Subscript/6tExBnkw">subscript</a> indicating the parent ring. Hence the examples above represent zero matrices over any ring. </p><p>The zero matrix also represents the <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformation</a> which sends all the <a href="/facts/Vector_(geometric)/bCLrdksy">vectors</a> to the <a href="/facts/Zero_vector/PvH2qhxh">zero vector</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> It is <a href="/facts/Idempotent_matrix/88Es7cEn">idempotent</a>, meaning that when it is multiplied by itself, the result is itself. </p><p>The zero matrix is the only matrix whose <a href="/facts/Rank_(linear_algebra)/Yp9b5LK3">rank</a> is 0. </p> <h2 id="occurrences">Occurrences</h2> <p>In <a href="/facts/Ordinary_least_squares/q7H9k5vM">ordinary least squares</a> regression, if there is a perfect fit to the data, the <a href="/facts/Annihilator_matrix/5bLbzjy9">annihilator matrix</a> is the zero matrix. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Identity_matrix/Glbcf6ol">Identity matrix</a>, the multiplicative identity for matrices</li> <li><a href="/facts/Matrix_of_ones/wvacr9uD">Matrix of ones</a>, a matrix where all elements are one</li> <li><a href="/facts/Nilpotent_matrix/qIns3FJJ">Nilpotent matrix</a></li> <li><a href="/facts/Single-entry_matrix/oH38TYtv">Single-entry matrix</a>, a matrix where all but one element is zero</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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. <a href="9780387964126" target="_blank">9780387964126</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"Intro to zero matrices (article) | Matrices". Khan Academy. Retrieved 2020-08-13. <a href="https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:properties-of-matrix-addition-and-scalar-multiplication/a/intro-to-zero-matrices" target="_blank">https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:properties-of-matrix-addition-and-scalar-multiplication/a/intro-to-zero-matrices</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Weisstein, Eric W. "Zero Matrix". mathworld.wolfram.com. Retrieved 2020-08-13. <a href="https://mathworld.wolfram.com/ZeroMatrix.html" target="_blank">https://mathworld.wolfram.com/ZeroMatrix.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>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. <a href="9780486663418" target="_blank">9780486663418</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>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. <a href="9780120887842" target="_blank">9780120887842</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>