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Reference.org
Vectorization (mathematics)
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, especially in <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a> and <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix theory</a>, the vectorization of a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> is a <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformation</a> which converts the matrix into a <a href="/facts/Vector_(mathematics_and_physics)/U8mM7A5F">vector</a>. Specifically, the vectorization of a <i>m</i> × <i>n</i> matrix <i>A</i>, denoted vec(<i>A</i>), is the <i>mn</i> × 1 column vector obtained by stacking the columns of the matrix <i>A</i> on top of one another: vec ( A ) = [ a 1 , 1 , … , a m , 1 , a 1 , 2 , … , a m , 2 , … , a 1 , n , … , a m , n ] T {\displaystyle \operatorname {vec} (A)=[a_{1,1},\ldots ,a_{m,1},a_{1,2},\ldots ,a_{m,2},\ldots ,a_{1,n},\ldots ,a_{m,n}]^{\mathrm {T} }} Here, a i , j {\displaystyle a_{i,j}} represents the element in the <i>i</i>-th row and <i>j</i>-th column of <i>A</i>, and the superscript T {\displaystyle {}^{\mathrm {T} }} denotes the <a href="/facts/Transpose/8wmsagGS">transpose</a>. Vectorization expresses, through coordinates, the <a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a> R m × n := R m ⊗ R n ≅ R m n {\displaystyle \mathbf {R} ^{m\times n}:=\mathbf {R} ^{m}\otimes \mathbf {R} ^{n}\cong \mathbf {R} ^{mn}} between these (i.e., of matrices and vectors) as vector spaces. </p><p>For example, for the 2×2 matrix A = [ a b c d ] {\displaystyle A={\begin{bmatrix}a&b\\c&d\end{bmatrix}}} , the vectorization is vec ( A ) = [ a c b d ] {\displaystyle \operatorname {vec} (A)={\begin{bmatrix}a\\c\\b\\d\end{bmatrix}}} . </p><p>The connection between the vectorization of <i>A</i> and the vectorization of its transpose is given by the <a href="/facts/Commutation_matrix/DHgJXDqy">commutation matrix</a>. </p>