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Matrix equivalence
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<h2 id="properties">Properties</h2> <p>Matrix equivalence is an <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a> on the space of rectangular matrices. </p><p>For two rectangular matrices of the same size, their equivalence can also be characterized by the following conditions </p> <ul><li>The matrices can be transformed into one another by a combination of <a href="/facts/Elementary_row_operation/coFCElwg">elementary row and column operations</a>.</li> <li>Two matrices are equivalent if and only if they have the same <a href="/facts/Rank_of_a_matrix/Yp9b5LK3">rank</a>.</li></ul> <p>If matrices are row equivalent then they are also matrix equivalent. However, the converse does not hold; matrices that are matrix equivalent are not necessarily row equivalent. This makes matrix equivalence a generalization of row equivalence.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> <h2 id="canonical-form">Canonical form</h2> <p>The <a href="/facts/Rank_of_a_matrix/Yp9b5LK3">rank</a> property yields an intuitive <a href="/facts/Canonical_form/K2jhB44k">canonical form</a> for matrices of the equivalence class of rank k {\displaystyle k} as </p><p> ( 1 0 0 ⋯ 0 0 1 0 ⋯ 0 0 0 ⋱ 0 ⋮ 1 ⋮ 0 ⋱ 0 ⋯ 0 ) {\displaystyle {\begin{pmatrix}1&0&0&&\cdots &&0\\0&1&0&&\cdots &&0\\0&0&\ddots &&&&0\\\vdots &&&1&&&\vdots \\&&&&0&&\\&&&&&\ddots &\\0&&&\cdots &&&0\end{pmatrix}}} , </p><p> where the number of 1 {\displaystyle 1} s on the diagonal is equal to k {\displaystyle k} . This is a special case of the <a href="/facts/Smith_normal_form/pchMkW7Q">Smith normal form</a>, which generalizes this concept on vector spaces to <a href="/facts/Free_module/o6rC6yoJ">free modules</a> over <a href="/facts/Principal_ideal_domain/JCwAUnyW">principal ideal domains</a>. Thus: </p><p>Theorem: Any <i>m</i>x<i>n</i> matrix of rank <i>k</i> is matrix equivalent to the <i>m</i>x<i>n</i> matrix that is all zeroes except that the first <i>k</i> diagonal entries are ones. <a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> Corollary: Matrix equivalent classes are characterized by rank: two same-sided matrices are matrix equivalent if and only if they have the same rank. <a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></p> <h2 id="2x2-matrices">2x2 matrices</h2> <p>2x2 matrices only have three possible ranks: zero, one, or two. This means all 2x2 matrices fit into one of three matrix equivalent classes:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p><p> ( 0 0 0 0 ) {\displaystyle {\begin{pmatrix}0&0\\0&0\\\end{pmatrix}}} , ( 1 0 0 0 ) {\displaystyle {\begin{pmatrix}1&0\\0&0\\\end{pmatrix}}} , ( 1 0 0 1 ) {\displaystyle {\begin{pmatrix}1&0\\0&1\\\end{pmatrix}}} </p><p>This means all 2x2 matrices are equivalent to one of these matrices. There is only one zero rank matrix, but the other two classes have infinitely many members; The representative matrices above are the simplest matrix for each class. </p> <h2 id="matrix-similarity">Matrix similarity</h2> <p class="note">Main article: <a href="/facts/Matrix_similarity/ySevpav1">Matrix similarity</a></p> <p>Matrix similarity is a special case of matrix equivalence. If two matrices are similar then they are also equivalent. However, the converse is not true.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> For example these two matrices are equivalent but not similar: </p><p> ( 1 0 0 1 ) {\displaystyle {\begin{pmatrix}1&0\\0&1\\\end{pmatrix}}} , ( 1 2 0 3 ) {\displaystyle {\begin{pmatrix}1&2\\0&3\\\end{pmatrix}}} </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Row_equivalence/UEhrr5gE">Row equivalence</a></li> <li><a href="/facts/Matrix_congruence/L5nPV4BQ">Matrix congruence</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Hefferon, Jim. Linear Algebra (4th ed.). pp. 270–272. This article incorporates text from this source, which is available under the CC BY-SA 3.0 license. <a href="https://hefferon.net/linearalgebra/" target="_blank">https://hefferon.net/linearalgebra/</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Hefferon, Jim. Linear Algebra (4th ed.). pp. 270–272. This article incorporates text from this source, which is available under the CC BY-SA 3.0 license. <a href="https://hefferon.net/linearalgebra/" target="_blank">https://hefferon.net/linearalgebra/</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Hefferon, Jim. Linear Algebra (4th ed.). pp. 270–272. This article incorporates text from this source, which is available under the CC BY-SA 3.0 license. <a href="https://hefferon.net/linearalgebra/" target="_blank">https://hefferon.net/linearalgebra/</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Hefferon, Jim. Linear Algebra (4th ed.). pp. 270–272. This article incorporates text from this source, which is available under the CC BY-SA 3.0 license. <a href="https://hefferon.net/linearalgebra/" target="_blank">https://hefferon.net/linearalgebra/</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Hefferon, Jim. Linear Algebra (4th ed.). p. 405. This article incorporates text from this source, which is available under the CC BY-SA 3.0 license. <a href="https://hefferon.net/linearalgebra/" target="_blank">https://hefferon.net/linearalgebra/</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>