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| In [[mathematics]], especially in [[linear algebra]] and [[Matrix (mathematics)|matrix theory]], the '''vectorization''' of a [[matrix (mathematics)|matrix]] is a [[linear transformation]] which converts the matrix into a [[column vector]]. Specifically, the vectorization of an ''m×n'' matrix ''A'', denoted by vec(''A''), is the ''mn × 1'' column vector obtained by stacking the columns of the matrix ''A'' on top of one another:
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| :<math>\mathrm{vec}(A) = [a_{1,1}, ..., a_{m,1}, a_{1,2}, ..., a_{m,2}, ..., a_{1,n}, ..., a_{m,n}]^T</math>
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| Here <math>a_{i,j}</math> represents the <math>(i,j)</math>-th element of matrix <math>A</math> and the superscript <math>^T</math> denotes the [[transpose]]. Vectorization expresses the [[isomorphism]] <math>\mathbf{R}^{m \times n} := \mathbf{R}^m \otimes \mathbf{R}^n \cong \mathbf{R}^{mn}</math> between these vector spaces (of matrices and vectors) in coordinates.
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| For example, for the 2×2 matrix <math>A</math> = <math>\begin{bmatrix} a & b \\ c & d \end{bmatrix}</math>, the vectorization is <math>\mathrm{vec}(A) = \begin{bmatrix} a \\ c \\ b \\ d \end{bmatrix}</math>.
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| ==Compatibility with Kronecker products==
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| The vectorization is frequently used together with the [[Kronecker product]] to express [[matrix multiplication]] as a linear transformation on matrices. In particular,
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| :<math> \mbox{vec}(ABC)=(C^{T}\otimes A)\mbox{vec}(B) </math>
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| for matrices ''A'', ''B'', and ''C'' of dimensions ''k×l'', ''l×m'', and ''m×n''. For example, if <math> \mbox{ad}_A(X) = AX-XA</math> (the [[adjoint endomorphism]] of the [[Lie algebra]] gl(''n'','''C''') of all ''n×n'' matrices with [[complex number|complex]] entries), then <math>\mbox{vec}(\mbox{ad}_A(X)) = (I_n\otimes A - A^T \otimes I_n ) \mbox{vec}(X)</math>, where <math>I_n</math> is the ''n×n'' [[identity matrix]].
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| There are two other useful formulations:
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| :<math> \mbox{vec}(ABC)=(I_n\otimes AB)\mbox{vec}(C) =(C^{T}B^{T}\otimes I_k)\mbox{vec}(A)</math>
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| :<math> \mbox{vec}(AB)=(I_m\otimes A)\mbox{vec}(B) =(B^{T}\otimes I_k)\mbox{vec}(A)</math>
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| ==Compatibility with Hadamard products==
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| Vectorization is an [[algebra homomorphism]] from the space of ''n×n'' matrices with the [[Hadamard product (matrices)|Hadamard]] (entrywise) product to '''C'''<sup>n</sup> with its [[Hadamard product]]{{dn|date=July 2013}}:
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| :vec(''A'' <math>\circ</math> ''B'') = vec(''A'') <math>\circ</math> vec(''B'').
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| ==Compatibility with inner products==
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| Vectorization is a [[unitary transformation]] from the space of ''n×n'' matrices with the [[Matrix norm#Frobenius norm|Frobenius]] (or [[Hilbert-Schmidt operator|Hilbert-Schmidt]]) [[inner product]] to '''C'''<sup>n</sup> :
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| :tr(''A''<sup>*</sup> ''B'') = vec(''A'')<sup>*</sup> vec(''B'')
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| where the superscript <sup>*</sup> denotes the [[conjugate transpose]].
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| ==Half-vectorization==
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| For a [[symmetric matrix]] ''A'', the vector vec(''A'') contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with the [[lower triangular matrix|lower triangular]] portion, that is, the ''n''(''n''+1)/2 entries on and below the [[main diagonal]]. For such matrices, the '''half-vectorization''' is sometimes more useful than the vectorization. The half-vectorization, vech(''A''), of a symmetric ''n×n'' matrix ''A'' is the ''n''(''n''+1)/2 × 1 column vector obtained by vectorizing only the lower triangular part of ''A'':
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| :vech(''A'') = [ ''A''<sub>1,1</sub>, ..., ''A''<sub>n,1</sub>, ''A''<sub>2,2</sub>, ..., ''A''<sub>n,2</sub>, ..., ''A''<sub>n-1,n-1</sub>,''A''<sub>n-1,n</sub>, ''A''<sub>n,n</sub> ]<sup>T</sup>.
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| For example, for the 2×2 matrix ''A'' = <math>\begin{bmatrix} a & b \\ b & d \end{bmatrix}</math>, the half-vectorization is vech(''A'') = <math>\begin{bmatrix} a \\ b \\ d \end{bmatrix}</math>.
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| There exist unique matrices transforming the half-vectorization of a matrix to its vectorization and vice-versa called, respectively, the [[duplication matrix]] and the [[elimination matrix]].
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| ==Programming language==
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| Programming languages that implement matrices may have easy means for vectorization.
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| In [[Matlab]]/[[GNU Octave]] a matrix <code>A</code> can be vectorized by <code>A(:)</code>.
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| In [[Python (programming language)|Python]] [[NumPy]] arrays implement the 'flatten' method (although this stacks the ''rows'' of the matrix, not the columns), while in [[R programming language|R]] the desired effect can be achieved via the 'c()' or 'as.vector()' functions.
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| ==See also==
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| * [[Voigt notation]]
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| * [[Row-major order|Column-major order]]
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| * [[Matricization]]
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| ==References==
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| *Jan R. Magnus and Heinz Neudecker (1999), ''Matrix Differential Calculus with Applications in Statistics and Econometrics'', 2nd Ed., Wiley. ISBN 0-471-98633-X.
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| *Jan R. Magnus (1988), ''Linear Structures'', Oxford University Press. ISBN 0-85264-299-7.
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| [[Category:Linear algebra]]
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| [[Category:Matrices]]
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