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Multilinear map

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</style><div role="note" class="hatnote navigation-not-searchable">For multilinear maps used in cryptography, see <a href="Cryptographic_multilinear_map" title="Cryptographic multilinear map">Cryptographic multilinear map</a>.</div>

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In <a href="Linear_algebra" title="Linear algebra">linear algebra</a>, a multilinear map is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> of several variables that is <a href="Linear_map" title="Linear map">linear</a> separately in each variable. More precisely, a multilinear map is a function

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<annotation encoding="application/x-tex">{\displaystyle f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}}</annotation>

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</math><img src="./a791997e08e880ad7aeb9808d71b559be172eeb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.396ex; height:2.509ex;" alt="{\displaystyle f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{1},\ldots ,V_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle V_{1},\ldots ,V_{n}}</annotation>

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</math><img src="./056aea51dd62880a078cf84fe69ff7d09afb20b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.161ex; height:2.509ex;" alt="{\displaystyle V_{1},\ldots ,V_{n}}" loading="lazy"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {Z} _{\geq 0}}">

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<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {Z} _{\geq 0}}</annotation>

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</math><img src="./95800ce5aede1cb3e0bd5908ec2290c0dd491d7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.118ex; height:2.676ex;" alt="{\displaystyle n\in \mathbb {Z} _{\geq 0}}" loading="lazy">) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

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</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> are <a href="Vector_space" title="Vector space">vector spaces</a> (or <a href="Module_(mathematics)" title="Module (mathematics)">modules</a> over a <a href="Commutative_ring" title="Commutative ring">commutative ring</a>), with the following property: for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">, if all of the variables but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{i}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle f(v_{1},\ldots ,v_{i},\ldots ,v_{n})}</annotation>

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</math><img src="./c39026502b25d6b385c4fde827d545482b76e073.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.9ex; height:2.843ex;" alt="{\displaystyle f(v_{1},\ldots ,v_{i},\ldots ,v_{n})}" loading="lazy"> is a <a href="Linear_map" title="Linear map">linear function</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{i}}</annotation>

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</math><img src="./7dffe5726650f6daac54829972a94f38eb8ec127.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i}}" loading="lazy">.<a href="#cite_note-1">[1]</a> One way to visualize this is to imagine two <a href="Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> vectors; if one of these vectors is scaled by a factor of 2 while the other remains unchanged, the <a href="Cross_product" title="Cross product">cross product</a> likewise scales by a factor of two. If both are scaled by a factor of 2, the cross product scales by a factor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle 2^{2}}</annotation>

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</math><img src="./efd7711cd907a2d46557a410fb67fc0d84c52ba3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 2^{2}}" loading="lazy">.

A multilinear map of one variable is a <a href="Linear_map" title="Linear map">linear map</a>, and of two variables is a <a href="Bilinear_map" title="Bilinear map">bilinear map</a>. More generally, for any nonnegative integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">, a multilinear map of k variables is called a k-linear map. If the <a href="Codomain" title="Codomain">codomain</a> of a multilinear map is the <a href="Field_of_scalars" class="mw-redirect" title="Field of scalars">field of scalars</a>, it is called a <a href="Multilinear_form" title="Multilinear form">multilinear form</a>. Multilinear maps and multilinear forms are fundamental objects of study in <a href="Multilinear_algebra" title="Multilinear algebra">multilinear algebra</a>.

If all variables belong to the same space, one can consider <a href="Symmetric_function" title="Symmetric function">symmetric</a>, <a href="Bilinear_form#Symmetric,_skew-symmetric_and_alternating_forms" title="Bilinear form">antisymmetric</a> and <a href="Alternating_map" class="mw-redirect" title="Alternating map">alternating</a> k-linear maps. The latter two coincide if the underlying <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> (or <a href="Field_(mathematics)" title="Field (mathematics)">field</a>) has a <a href="Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> different from two, else the former two coincide.

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<ul><li>Any <a href="Bilinear_map" title="Bilinear map">bilinear map</a> is a multilinear map. For example, any <a href="Inner_product" class="mw-redirect" title="Inner product">inner product</a> on a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">-vector space is a multilinear map, as is the <a href="Cross_product" title="Cross product">cross product</a> of vectors in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}">

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</math><img src="./f936ddf584f8f3dd2a0ed08917001b7a404c10b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" loading="lazy">.</li>

<li>The <a href="Determinant" title="Determinant">determinant</a> of a <a href="Square_matrix" title="Square matrix">square matrix</a> is a multilinear function of the columns (or rows); it is also an <a href="Alternating_form" class="mw-redirect" title="Alternating form">alternating</a> function of the columns (or rows).</li>

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<annotation encoding="application/x-tex">{\displaystyle F\colon \mathbb {R} ^{m}\to \mathbb {R} ^{n}}</annotation>

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</math><img src="./14253f40a74fe32b7e39018f2432fd660cad6763.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.638ex; height:2.343ex;" alt="{\displaystyle F\colon \mathbb {R} ^{m}\to \mathbb {R} ^{n}}" loading="lazy"> is a <a href="Smooth_function" class="mw-redirect" title="Smooth function">Ck function</a>, then the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

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</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> at each point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> in its domain can be viewed as a <a href="Symmetric_function" title="Symmetric function">symmetric</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-linear function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{k}\!F\colon \mathbb {R} ^{m}\times \cdots \times \mathbb {R} ^{m}\to \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle D^{k}\!F\colon \mathbb {R} ^{m}\times \cdots \times \mathbb {R} ^{m}\to \mathbb {R} ^{n}}</annotation>

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</math><img src="./e91a25266b97ec183498264455f4e6e7f34ee205.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:27.021ex; height:2.676ex;" alt="{\displaystyle D^{k}\!F\colon \mathbb {R} ^{m}\times \cdots \times \mathbb {R} ^{m}\to \mathbb {R} ^{n}}" loading="lazy">.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Coordinate_representation">Coordinate representation</h2></div>

Let

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<annotation encoding="application/x-tex">{\displaystyle f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}}</annotation>

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be a multilinear map between <a href="Finite-dimensional" class="mw-redirect" title="Finite-dimensional">finite-dimensional</a> vector spaces, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}\!}">

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<annotation encoding="application/x-tex">{\displaystyle V_{i}\!}</annotation>

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</math><img src="./2fc5bf0d79e347d23ec65efb17b7b053473965d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle V_{i}\!}" loading="lazy"> has dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{i}\!}">

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<annotation encoding="application/x-tex">{\displaystyle d_{i}\!}</annotation>

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</math><img src="./c9a3a18594aa00be3cd484cece489974dbbcf991.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:2.009ex; height:2.509ex;" alt="{\displaystyle d_{i}\!}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W\!}">

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<annotation encoding="application/x-tex">{\displaystyle W\!}</annotation>

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</math><img src="./00717b7c932d47b8051bbf36f279fed6a2399d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W\!}" loading="lazy"> has dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\!}">

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<annotation encoding="application/x-tex">{\displaystyle d\!}</annotation>

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</math><img src="./97a0678efcce4287c9baf3382400224f7da1c447.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d\!}" loading="lazy">. If we choose a <a href="Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\textbf {e}}_{i1},\ldots ,{\textbf {e}}_{id_{i}}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{{\textbf {e}}_{i1},\ldots ,{\textbf {e}}_{id_{i}}\}}</annotation>

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</math><img src="./c48d88eab59c6b4b136f55787e947a15c056be61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.855ex; height:3.009ex;" alt="{\displaystyle \{{\textbf {e}}_{i1},\ldots ,{\textbf {e}}_{id_{i}}\}}" loading="lazy"> for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}\!}">

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<annotation encoding="application/x-tex">{\displaystyle V_{i}\!}</annotation>

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</math><img src="./2fc5bf0d79e347d23ec65efb17b7b053473965d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle V_{i}\!}" loading="lazy"> and a basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\textbf {b}}_{1},\ldots ,{\textbf {b}}_{d}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{{\textbf {b}}_{1},\ldots ,{\textbf {b}}_{d}\}}</annotation>

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</math><img src="./9a4b0ec7dd7cb7f26987a71d7b00b24b2876c82a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.62ex; height:2.843ex;" alt="{\displaystyle \{{\textbf {b}}_{1},\ldots ,{\textbf {b}}_{d}\}}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W\!}">

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<annotation encoding="application/x-tex">{\displaystyle W\!}</annotation>

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</math><img src="./00717b7c932d47b8051bbf36f279fed6a2399d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W\!}" loading="lazy"> (using bold for vectors), then we can define a collection of scalars <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{j_{1}\cdots j_{n}}^{k}}">

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<annotation encoding="application/x-tex">{\displaystyle A_{j_{1}\cdots j_{n}}^{k}}</annotation>

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</math><img src="./5123222b1c0555d64b8348aad5baa17e4d34c160.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.052ex; height:3.509ex;" alt="{\displaystyle A_{j_{1}\cdots j_{n}}^{k}}" loading="lazy"> by

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<annotation encoding="application/x-tex">{\displaystyle f({\textbf {e}}_{1j_{1}},\ldots ,{\textbf {e}}_{nj_{n}})=A_{j_{1}\cdots j_{n}}^{1}\,{\textbf {b}}_{1}+\cdots +A_{j_{1}\cdots j_{n}}^{d}\,{\textbf {b}}_{d}.}</annotation>

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</math><img src="./5d07690f959c7d0838656d909feb5ed22befd93b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:48.286ex; height:3.509ex;" alt="{\displaystyle f({\textbf {e}}_{1j_{1}},\ldots ,{\textbf {e}}_{nj_{n}})=A_{j_{1}\cdots j_{n}}^{1}\,{\textbf {b}}_{1}+\cdots +A_{j_{1}\cdots j_{n}}^{d}\,{\textbf {b}}_{d}.}" loading="lazy"></dd></dl>

Then the scalars <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{A_{j_{1}\cdots j_{n}}^{k}\mid 1\leq j_{i}\leq d_{i},1\leq k\leq d\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{A_{j_{1}\cdots j_{n}}^{k}\mid 1\leq j_{i}\leq d_{i},1\leq k\leq d\}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\textbf {v}}_{i}=\sum _{j=1}^{d_{i}}v_{ij}{\textbf {e}}_{ij}\!}</annotation>

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</math><img src="./45610ff4e420302a03b9b0d7da8e8763ad72d9be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-right: -0.387ex; width:14.358ex; height:7.843ex;" alt="{\displaystyle {\textbf {v}}_{i}=\sum _{j=1}^{d_{i}}v_{ij}{\textbf {e}}_{ij}\!}" loading="lazy"></dd></dl>

for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq i\leq n\!}">

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<annotation encoding="application/x-tex">{\displaystyle f({\textbf {v}}_{1},\ldots ,{\textbf {v}}_{n})=\sum _{j_{1}=1}^{d_{1}}\cdots \sum _{j_{n}=1}^{d_{n}}\sum _{k=1}^{d}A_{j_{1}\cdots j_{n}}^{k}v_{1j_{1}}\cdots v_{nj_{n}}{\textbf {b}}_{k}.}</annotation>

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</math><img src="./1aa5bcb30ebd68e758fd61bebf5afc7805670ae7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:52.888ex; height:7.843ex;" alt="{\displaystyle f({\textbf {v}}_{1},\ldots ,{\textbf {v}}_{n})=\sum _{j_{1}=1}^{d_{1}}\cdots \sum _{j_{n}=1}^{d_{n}}\sum _{k=1}^{d}A_{j_{1}\cdots j_{n}}^{k}v_{1j_{1}}\cdots v_{nj_{n}}{\textbf {b}}_{k}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Example">Example</h2></div>

Let's take a trilinear function

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<annotation encoding="application/x-tex">{\displaystyle g\colon R^{2}\times R^{2}\times R^{2}\to R,}</annotation>

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where Vi = R2, di = 2, i = 1,2,3, and W = R, d = 1.

A basis for each Vi is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\textbf {e}}_{i1},\ldots ,{\textbf {e}}_{id_{i}}\}=\{{\textbf {e}}_{1},{\textbf {e}}_{2}\}=\{(1,0),(0,1)\}.}">

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<annotation encoding="application/x-tex">{\displaystyle \{{\textbf {e}}_{i1},\ldots ,{\textbf {e}}_{id_{i}}\}=\{{\textbf {e}}_{1},{\textbf {e}}_{2}\}=\{(1,0),(0,1)\}.}</annotation>

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</math><img src="./8eb7fedacde3c961cee4e0f0b65d7b49f5905f39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:42.311ex; height:3.009ex;" alt="{\displaystyle \{{\textbf {e}}_{i1},\ldots ,{\textbf {e}}_{id_{i}}\}=\{{\textbf {e}}_{1},{\textbf {e}}_{2}\}=\{(1,0),(0,1)\}.}" loading="lazy"> Let

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<annotation encoding="application/x-tex">{\displaystyle g({\textbf {e}}_{1i},{\textbf {e}}_{2j},{\textbf {e}}_{3k})=f({\textbf {e}}_{i},{\textbf {e}}_{j},{\textbf {e}}_{k})=A_{ijk},}</annotation>

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</math><img src="./4bf1d1272bc7ab6c71a041888f359e47319fc566.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.483ex; height:3.009ex;" alt="{\displaystyle g({\textbf {e}}_{1i},{\textbf {e}}_{2j},{\textbf {e}}_{3k})=f({\textbf {e}}_{i},{\textbf {e}}_{j},{\textbf {e}}_{k})=A_{ijk},}" loading="lazy"></dd></dl>

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</math><img src="./9e22cd7ef6de1c6ba52f9f4a3866dae37fc31634.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.564ex; height:2.843ex;" alt="{\displaystyle i,j,k\in \{1,2\}}" loading="lazy">. In other words, the constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{ijk}}">

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</math><img src="./fdab6c0fd20e179f83bb20f6a76768bb26c30380.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.077ex; height:2.843ex;" alt="{\displaystyle A_{ijk}}" loading="lazy"> is a function value at one of the eight possible triples of basis vectors (since there are two choices for each of the three <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle V_{i}}</annotation>

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</math><img src="./f300b83673e961a9d48f3862216b167f94e5668c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle V_{i}}" loading="lazy">), namely:

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<annotation encoding="application/x-tex">{\displaystyle \{{\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{1}\},\{{\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{2}\},\{{\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{1}\},\{{\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{2}\},\{{\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{1}\},\{{\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{2}\},\{{\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{1}\},\{{\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{2}\}.}</annotation>

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</math><img src="./5e980321c63bc49ae91cfeee9e9dc6a63e574eec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:97.732ex; height:2.843ex;" alt="{\displaystyle \{{\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{1}\},\{{\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{2}\},\{{\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{1}\},\{{\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{2}\},\{{\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{1}\},\{{\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{2}\},\{{\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{1}\},\{{\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{2}\}.}" loading="lazy"></dd></dl>

Each vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {v}}_{i}\in V_{i}=R^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\textbf {v}}_{i}\in V_{i}=R^{2}}</annotation>

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</math><img src="./c1afa94a416faeb8359eda6b9bd1da5feffb29f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.123ex; height:3.009ex;" alt="{\displaystyle {\textbf {v}}_{i}\in V_{i}=R^{2}}" loading="lazy"> can be expressed as a <a href="Linear_combination" title="Linear combination">linear combination</a> of the basis vectors

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<annotation encoding="application/x-tex">{\displaystyle {\textbf {v}}_{i}=\sum _{j=1}^{2}v_{ij}{\textbf {e}}_{ij}=v_{i1}\times {\textbf {e}}_{1}+v_{i2}\times {\textbf {e}}_{2}=v_{i1}\times (1,0)+v_{i2}\times (0,1).}</annotation>

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</math><img src="./195f76aa1c3e82031ab1dac34551d44a84b4209a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:64.137ex; height:7.676ex;" alt="{\displaystyle {\textbf {v}}_{i}=\sum _{j=1}^{2}v_{ij}{\textbf {e}}_{ij}=v_{i1}\times {\textbf {e}}_{1}+v_{i2}\times {\textbf {e}}_{2}=v_{i1}\times (1,0)+v_{i2}\times (0,1).}" loading="lazy"></dd></dl>

The function value at an arbitrary collection of three vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {v}}_{i}\in R^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\textbf {v}}_{i}\in R^{2}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle g({\textbf {v}}_{1},{\textbf {v}}_{2},{\textbf {v}}_{3})=\sum _{i=1}^{2}\sum _{j=1}^{2}\sum _{k=1}^{2}A_{ijk}v_{1i}v_{2j}v_{3k},}</annotation>

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</math><img src="./00c3c726ba2215d8d95f0ab629653e6464e2fc54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:40.084ex; height:7.676ex;" alt="{\displaystyle g({\textbf {v}}_{1},{\textbf {v}}_{2},{\textbf {v}}_{3})=\sum _{i=1}^{2}\sum _{j=1}^{2}\sum _{k=1}^{2}A_{ijk}v_{1i}v_{2j}v_{3k},}" loading="lazy"></dd></dl>

or in expanded form as

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g((a,b),(c,d)&,(e,f))=ace\times g({\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{1})+acf\times g({\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{2})\\&+ade\times g({\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{1})+adf\times g({\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{2})+bce\times g({\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{1})+bcf\times g({\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{2})\\&+bde\times g({\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{1})+bdf\times g({\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{2}).\end{aligned}}}</annotation>

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</math><img src="./8004aa7c471298597489932310d46c814df7592e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:97.616ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}g((a,b),(c,d)&,(e,f))=ace\times g({\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{1})+acf\times g({\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{2})\\&+ade\times g({\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{1})+adf\times g({\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{2})+bce\times g({\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{1})+bcf\times g({\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{2})\\&+bde\times g({\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{1})+bdf\times g({\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{2}).\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Relation_to_tensor_products">Relation to tensor products</h2></div>

There is a <a href="Natural_transformation" title="Natural transformation">natural</a> one-to-one correspondence between multilinear maps

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and linear maps

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<annotation encoding="application/x-tex">{\displaystyle f(v_{1},\ldots ,v_{n})=F(v_{1}\otimes \cdots \otimes v_{n}).}</annotation>

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</math><img src="./fabd5694e57568da60d1396aeabc37bfe518679d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.021ex; height:2.843ex;" alt="{\displaystyle f(v_{1},\ldots ,v_{n})=F(v_{1}\otimes \cdots \otimes v_{n}).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Multilinear_functions_on_n×n_matrices">Multilinear functions on n×n matrices</h2></div>

One can consider multilinear functions, on an n×n matrix over a <a href="Commutative_ring" title="Commutative ring">commutative ring</a> K with identity, as a function of the rows (or equivalently the columns) of the matrix. Let A be such a matrix and ai, 1 ≤ i ≤ n, be the rows of A. Then the multilinear function D can be written as

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<annotation encoding="application/x-tex">{\displaystyle D(A)=D(a_{1},\ldots ,a_{n}),}</annotation>

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</math><img src="./2d95ff355759d90567c94bab52d2be90b17044b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.866ex; height:2.843ex;" alt="{\displaystyle D(A)=D(a_{1},\ldots ,a_{n}),}" loading="lazy"></dd></dl>

satisfying

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<annotation encoding="application/x-tex">{\displaystyle D(a_{1},\ldots ,ca_{i}+a_{i}',\ldots ,a_{n})=cD(a_{1},\ldots ,a_{i},\ldots ,a_{n})+D(a_{1},\ldots ,a_{i}',\ldots ,a_{n}).}</annotation>

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</math><img src="./2ba3bf8e7663ef1cf66d92364e8cb4f057842baa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:76.025ex; height:3.009ex;" alt="{\displaystyle D(a_{1},\ldots ,ca_{i}+a_{i}',\ldots ,a_{n})=cD(a_{1},\ldots ,a_{i},\ldots ,a_{n})+D(a_{1},\ldots ,a_{i}',\ldots ,a_{n}).}" loading="lazy"></dd></dl>

If we let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {e}}_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {e}}_{j}}</annotation>

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</math><img src="./267ebc38efac55c78d8d73675f1e2740776cb83d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.201ex; height:2.843ex;" alt="{\displaystyle {\hat {e}}_{j}}" loading="lazy"> represent the jth row of the <a href="Identity_matrix" title="Identity matrix">identity matrix</a>, we can express each row ai as the sum

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<annotation encoding="application/x-tex">{\displaystyle a_{i}=\sum _{j=1}^{n}A(i,j){\hat {e}}_{j}.}</annotation>

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</math><img src="./a432b35b0942983d76d9c97ae6f1eb2b39979b3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18.065ex; height:7.176ex;" alt="{\displaystyle a_{i}=\sum _{j=1}^{n}A(i,j){\hat {e}}_{j}.}" loading="lazy"></dd></dl>

Using the multilinearity of D we rewrite D(A) as

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<annotation encoding="application/x-tex">{\displaystyle D(A)=D\left(\sum _{j=1}^{n}A(1,j){\hat {e}}_{j},a_{2},\ldots ,a_{n}\right)=\sum _{j=1}^{n}A(1,j)D({\hat {e}}_{j},a_{2},\ldots ,a_{n}).}</annotation>

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</math><img src="./d9edbeff3909bda7958df41684d30993bff6ecd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:69.236ex; height:7.676ex;" alt="{\displaystyle D(A)=D\left(\sum _{j=1}^{n}A(1,j){\hat {e}}_{j},a_{2},\ldots ,a_{n}\right)=\sum _{j=1}^{n}A(1,j)D({\hat {e}}_{j},a_{2},\ldots ,a_{n}).}" loading="lazy"></dd></dl>

Continuing this substitution for each ai we get, for 1 ≤ i ≤ n,

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<annotation encoding="application/x-tex">{\displaystyle D(A)=\sum _{1\leq k_{1}\leq n}\ldots \sum _{1\leq k_{i}\leq n}\ldots \sum _{1\leq k_{n}\leq n}A(1,k_{1})A(2,k_{2})\dots A(n,k_{n})D({\hat {e}}_{k_{1}},\dots ,{\hat {e}}_{k_{n}}).}</annotation>

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</math><img src="./c0ade79188714293a0c31aafe19cde3d1500be20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:78.096ex; height:5.843ex;" alt="{\displaystyle D(A)=\sum _{1\leq k_{1}\leq n}\ldots \sum _{1\leq k_{i}\leq n}\ldots \sum _{1\leq k_{n}\leq n}A(1,k_{1})A(2,k_{2})\dots A(n,k_{n})D({\hat {e}}_{k_{1}},\dots ,{\hat {e}}_{k_{n}}).}" loading="lazy"></dd></dl>

Therefore, D(A) is uniquely determined by how D operates on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {e}}_{k_{1}},\dots ,{\hat {e}}_{k_{n}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {e}}_{k_{1}},\dots ,{\hat {e}}_{k_{n}}}</annotation>

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</math><img src="./a9d0edbeb7ec75efdae46f22818ae9524aa3011f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.735ex; height:2.843ex;" alt="{\displaystyle {\hat {e}}_{k_{1}},\dots ,{\hat {e}}_{k_{n}}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Example_2">Example</h2></div>

In the case of 2×2 matrices, we get

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<annotation encoding="application/x-tex">{\displaystyle D(A)=A_{1,1}A_{1,2}D({\hat {e}}_{1},{\hat {e}}_{1})+A_{1,1}A_{2,2}D({\hat {e}}_{1},{\hat {e}}_{2})+A_{1,2}A_{2,1}D({\hat {e}}_{2},{\hat {e}}_{1})+A_{1,2}A_{2,2}D({\hat {e}}_{2},{\hat {e}}_{2}),\,}</annotation>

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</math><img src="./6e6ebd37727c0da7d3a199537f648bc4a5d712a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:88.581ex; height:3.009ex;" alt="{\displaystyle D(A)=A_{1,1}A_{1,2}D({\hat {e}}_{1},{\hat {e}}_{1})+A_{1,1}A_{2,2}D({\hat {e}}_{1},{\hat {e}}_{2})+A_{1,2}A_{2,1}D({\hat {e}}_{2},{\hat {e}}_{1})+A_{1,2}A_{2,2}D({\hat {e}}_{2},{\hat {e}}_{2}),\,}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {e}}_{1}=[1,0]}</annotation>

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</math><img src="./bdd794a3255c63d344516713c9504fcd3bb4ebcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.097ex; height:2.843ex;" alt="{\displaystyle {\hat {e}}_{1}=[1,0]}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {e}}_{2}=[0,1]}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {e}}_{2}=[0,1]}</annotation>

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</math><img src="./63b4fcea12fab3c0b76c1e6e7eecb11be4c0394c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.097ex; height:2.843ex;" alt="{\displaystyle {\hat {e}}_{2}=[0,1]}" loading="lazy">. If we restrict <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

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</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> to be an alternating function, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D({\hat {e}}_{1},{\hat {e}}_{1})=D({\hat {e}}_{2},{\hat {e}}_{2})=0}">

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<annotation encoding="application/x-tex">{\displaystyle D({\hat {e}}_{1},{\hat {e}}_{1})=D({\hat {e}}_{2},{\hat {e}}_{2})=0}</annotation>

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</math><img src="./98011b60b22b561de99ccd95d1dbb7162c1608e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.278ex; height:2.843ex;" alt="{\displaystyle D({\hat {e}}_{1},{\hat {e}}_{1})=D({\hat {e}}_{2},{\hat {e}}_{2})=0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D({\hat {e}}_{2},{\hat {e}}_{1})=-D({\hat {e}}_{1},{\hat {e}}_{2})=-D(I)}">

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<annotation encoding="application/x-tex">{\displaystyle D({\hat {e}}_{2},{\hat {e}}_{1})=-D({\hat {e}}_{1},{\hat {e}}_{2})=-D(I)}</annotation>

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</math><img src="./25d0d8c7d13cbd41f6c16280b40715c50742f720.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.637ex; height:2.843ex;" alt="{\displaystyle D({\hat {e}}_{2},{\hat {e}}_{1})=-D({\hat {e}}_{1},{\hat {e}}_{2})=-D(I)}" loading="lazy">. Letting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(I)=1}">

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<annotation encoding="application/x-tex">{\displaystyle D(I)=1}</annotation>

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</math><img src="./a83d36a8a65537421e07c9750aa02755d9daf604.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.166ex; height:2.843ex;" alt="{\displaystyle D(I)=1}" loading="lazy">, we get the determinant function on 2×2 matrices:

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<annotation encoding="application/x-tex">{\displaystyle D(A)=A_{1,1}A_{2,2}-A_{1,2}A_{2,1}.}</annotation>

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</math><img src="./73d5acaff8f64e103de11afcd45b0135351ae5ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.369ex; height:3.009ex;" alt="{\displaystyle D(A)=A_{1,1}A_{2,2}-A_{1,2}A_{2,1}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<ul><li>A multilinear map has a value of zero whenever one of its arguments is zero.</li></ul>

<ul><li><a href="Algebraic_form" class="mw-redirect" title="Algebraic form">Algebraic form</a></li>

<li><a href="Multilinear_form" title="Multilinear form">Multilinear form</a></li>

<li><a href="Homogeneous_polynomial" title="Homogeneous polynomial">Homogeneous polynomial</a></li>

<li><a href="Homogeneous_function" title="Homogeneous function">Homogeneous function</a></li>

<li><a href="Tensor" title="Tensor">Tensors</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFLang2005" class="citation book cs1"><a href="Serge_Lang" title="Serge Lang">Lang, Serge</a> (2005) [2002]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Fge-BwqhqIYC&pg=PA511">"XIII. Matrices and Linear Maps §S Determinants"</a>. Algebra. Graduate Texts in Mathematics. Vol. 211 (3rd ed.). Springer. pp. 511–. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-95385-4</bdi>.</cite>

</li>

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