Matrix representation

<h2 id="basic-mathematical-operations">Basic mathematical operations</h2>
<p class="note">Main article: <a href="/facts/Matrix_(mathematics)/qa8FI4ko">Matrix (mathematics) § Basic operations</a></p>
<p>An m × n (read as m by n) order <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> is a set of numbers arranged in m rows and n columns. Matrices of the same order can be added by adding the corresponding elements. Two matrices can be multiplied, the condition being that the number of columns of the first matrix is equal to the number of rows of the second matrix. Hence, if an m × n matrix is multiplied with an n × r matrix, then the resultant matrix will be of the order m × r.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p><p>Operations like row operations or column operations can be performed on a matrix, using which we can obtain the inverse of a matrix. The inverse may be obtained by determining the adjoint as well.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> rows and columns are the different classes of matrices
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<h2 id="in-3d-graphics">In 3D graphics</h2>
<p>The choice of representation for 4×4 matrices commonly used in <a href="/facts/3D_graphics/zlWW6eaF">3D graphics</a> affects the implementation of matrix/vector operations in systems with packed <a href="/facts/SIMD_instruction/8p8vwUeE">SIMD instructions</a>:
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<h3>Row major (SoA)</h3>
<p>With row-major matrix order, it is easy to transform vectors using <a href="/facts/Dot_product/tNz8MLjT">dot product</a> operations, since the coefficients of each component are sequential in memory. Consequently, this layout may be desirable if a processor supports dot product operations natively. It is also possible to efficiently use a '3×4' affine transformation matrix without padding or awkward permutes.
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<h3>Column major (AoS)</h3>
<p>With column-major order, a "matrix × vector" multiply can be implemented with vectorized <a href="/facts/Multiply-add/L26C9A1Y">multiply-add</a> operations, if the vector's components are broadcast to each <a href="/facts/SIMD_lane/kRaRt8NX">SIMD lane</a>. It is also easy to access the <a href="/facts/Basis_vector/89IPoN6c">basis vectors</a> represented by a <a href="/facts/Transformation_matrix/qAwPXmpR">transformation matrix</a> as individual column vectors, as these are contiguous in memory.
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<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Row-_and_column-major_order/eVYvNWoE">Row- and column-major order</a></li>
<li><a href="/facts/Sparse_matrix/FFtN0XjR">Sparse matrix</a></li>
<li><a href="/facts/Skyline_matrix/7Tyzxcvg">Skyline matrix</a></li>
<li><a href="/facts/Locality_of_reference/fYFKDv5q">Locality of reference</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="http://developer.r-project.org/Sparse.html">a description of sparse matrices</a> in R.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>"Representation of Orthogonal or Unitary Matrices". University of Texas at Austin. Retrieved 14 September 2011. <a href="http://www.ma.utexas.edu/documentation/lapack/node117.html" target="_blank">http://www.ma.utexas.edu/documentation/lapack/node117.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Lehoucq, R. (1996). "The Computation of Elementary Unitary Matrices". ACM Transactions on Mathematical Software. 22 (4): 393–400. doi:10.1145/235815.235817. hdl:1911/101830. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Ramana, B.V (2008). Higher Engineering Mathematics. New Delhi: Tata Mcgraw-Hill. ISBN 978-0-07-063419-0. <a href="978-0-07-063419-0" target="_blank">978-0-07-063419-0</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Ramana, B.V (2008). Higher Engineering Mathematics. New Delhi: Tata Mcgraw-Hill. ISBN 978-0-07-063419-0. <a href="978-0-07-063419-0" target="_blank">978-0-07-063419-0</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Matrix representation open-in-new

Matrix representation