Inmathematics, especially inlinear algebra andmatrix theory, thevectorization of amatrix is alinear transformation which converts the matrix into avector. Specifically, the vectorization of am ×n matrixA, denoted vec(A), is themn × 1 column vector obtained by stacking the columns of the matrixA on top of one another:$${\displaystyle \mathrm{vec}(A)=[a_{1,1},\dots ,a_{m,1},a_{1,2},\dots ,a_{m,2},\dots ,a_{1,n},\dots ,a_{m,n}{]}^{\mathrm{T}}}$$Here,${\displaystyle a_{i,j}}$ represents the element in thei-th row andj-th column ofA, and the superscript${\displaystyle {}^{\mathrm{T}}}$ denotes thetranspose. Vectorization expresses, through coordinates, theisomorphism${\displaystyle {\mathbf{R}}^{m\times n}:={\mathbf{R}}^m\otimes {\mathbf{R}}^n\cong {\mathbf{R}}^{mn}}$ between these (i.e., of matrices and vectors) as vector spaces.

For example, for the 2×2 matrix, the vectorization is${\displaystyle \mathrm{vec}(A)=\left[\begin{array}{c}a\\ c\\ b\\ d\end{array}\right]}$.

The connection between the vectorization ofA and the vectorization of its transpose is given by thecommutation matrix.

The vectorization is frequently used together with theKronecker product to expressmatrix multiplication as a linear transformation on matrices. In particular,$${\displaystyle \mathrm{vec}(ABC)=(C^{\mathrm{T}}\otimes A)\mathrm{vec}(B)}$$for matricesA,B, andC of dimensionsk×l,l×m, andm×n.^{[note 1]} For example, if${\displaystyle {\mathrm{ad}}_A(X)=AX-XA}$ (theadjoint endomorphism of theLie algebragl(n,C) of alln×n matrices withcomplex entries), then${\displaystyle \mathrm{vec}({\mathrm{ad}}_A(X))=(A\otimes I_n-I_n\otimes A^{\mathrm{T}})\text{vec}(X)}$, where${\displaystyle I_n}$ is then×nidentity matrix.

There are two other useful formulations:

IfB is adiagonal matrix (i.e.,$B=\mathrm{diag}(b_1,\dots ,b_n)$), the vectorization can be written using the column-wise Kronecker product$\ast$ (seeKhatri-Rao product) and themain diagonal$b={\left[\begin{array}{c}{b}_1,\dots ,b_n\end{array}\right]}^{\mathrm{T}}$ ofB:$${\displaystyle \mathrm{vec}(ABC)=(C^{\mathrm{T}}\ast A)b}$$

More generally, it has been shown that vectorization is aself-adjunction in the monoidal closed structure of any category of matrices.^[1]

Vectorization is analgebra homomorphism from the space ofn ×n matrices with theHadamard (entrywise) product toCⁿ² with its Hadamard product:$${\displaystyle \mathrm{vec}(A\circ B)=\mathrm{vec}(A)\circ \mathrm{vec}(B).}$$

Vectorization is aunitary transformation from the space ofn×n matrices with theFrobenius (orHilbert–Schmidt)inner product toCⁿ²:$${\displaystyle \mathrm{tr}(A^{†}B)=\mathrm{vec}(A{)}^{†}\mathrm{vec}(B),}$$where the superscript^† denotes theconjugate transpose.

The matrix vectorization operation can be written in terms of a linear sum. LetX be anm ×n matrix that we want to vectorize, and lete_i be thei-th canonical basis vector for then-dimensional space, that is${\mathbf{e}}_i={\left[0,\dots ,0,1,0,\dots ,0\right]}^{\mathrm{T}}$. LetB_i be a(mn) ×m block matrix defined as follows:$${\displaystyle {\mathbf{B}}_i=\left[\begin{array}{c}\mathbf{0}\\ ⋮\\ \mathbf{0}\\ {\mathbf{I}}_m\\ \mathbf{0}\\ ⋮\\ \mathbf{0}\end{array}\right]={\mathbf{e}}_i\otimes {\mathbf{I}}_m}$$

B_i consists ofn block matrices of sizem ×m, stacked column-wise, and all these matrices are all-zero except for thei-th one, which is am ×m identity matrixI_m.

Then the vectorized version ofX can be expressed as follows:$${\displaystyle \mathrm{vec}(\mathbf{X})=\sum _{i=1}^n{\mathbf{B}}_i\mathbf{X}{\mathbf{e}}_i}$$

Multiplication ofX bye_i extracts thei-th column, while multiplication byB_i puts it into the desired position in the final vector.

Alternatively, the linear sum can be expressed using theKronecker product:$${\displaystyle \mathrm{vec}(\mathbf{X})=\sum _{i=1}^n{\mathbf{e}}_i\otimes \mathbf{X}{\mathbf{e}}_i}$$

For asymmetric matrixA, the vector vec(A) contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with thelower triangular portion, that is, then(n + 1)/2 entries on and below themain diagonal. For such matrices, thehalf-vectorization is sometimes more useful than the vectorization. The half-vectorization, vech(A), of a symmetricn ×n matrixA is then(n + 1)/2 × 1 column vector obtained by vectorizing only the lower triangular part ofA:$${\displaystyle \mathrm{vech}(A)=[A_{1,1},\dots ,A_{n,1},A_{2,2},\dots ,A_{n,2},\dots ,A_{n-1,n-1},A_{n,n-1},A_{n,n}{]}^{\mathrm{T}}.}$$

For example, for the 2×2 matrix, the half-vectorization is${\displaystyle \mathrm{vech}(A)=\left[\begin{array}{c}a\\ b\\ d\end{array}\right]}$.

There exist unique matrices transforming the half-vectorization of a matrix to its vectorization and vice versa called, respectively, theduplication matrix and theelimination matrix.

Programming languages that implement matrices may have easy means for vectorization.InMatlab/GNU Octave a matrixA can be vectorized byA(:).GNU Octave also allows vectorization and half-vectorization withvec(A) andvech(A) respectively.Julia has thevec(A) function as well.InPythonNumPy arrays implement theflatten method,^{[note 1]} while inR the desired effect can be achieved via thec() oras.vector() functions or, more efficiently, by removing the dimensions attribute of a matrixA withdim(A) <- NULL. InR, functionvec() of package 'ks' allows vectorization and functionvech() implemented in both packages 'ks' and 'sn' allows half-vectorization.^[2][3][4]

Vectorization is used inmatrix calculus and its applications in establishing e.g., moments of random vectors and matrices, asymptotics, as well as Jacobian and Hessian matrices.^[5]It is also used in local sensitivity and statistical diagnostics.^[6]

• Duplication and elimination matrices
• Voigt notation
• Packed storage matrix
• Column-major order
• Matricization

• Linear algebra
• Matrices (mathematics)

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