In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a vector. Specifically, the vectorization of a m × n matrix A, denoted vec(A), is the mn × 1 column vector obtained by stacking the columns of the matrix A 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 ] ⊤ {\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 {\top } }}
Here, a i , j {\displaystyle a_{i,j}} represents the element in the i-th row and j-th column of A, and the superscript
⊤ {\displaystyle {}^{\mathrm {\top } }} denotes the transpose. In other words, vec(A) is a vector containing the entries of A in column-major order. Vectorization expresses, through coordinates, the isomorphism R m × n ≅ R m n {\displaystyle \mathbf {R} ^{m\times n}\cong \mathbf {R} ^{mn}} between these (i.e., of matrices and vectors) as vector spaces. 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}}} . The connection between the vectorization of A and the vectorization of its transpose is given by the commutation matrix.
Compatibility with Kronecker products The vectorization is frequently used together with the Kronecker product to express matrix multiplication as a linear transformation on matrices. In particular,
vec ( A B C ) = ( C ⊤ ⊗ A ) vec ( B ) {\displaystyle \operatorname {vec} (ABC)=(C^{\mathrm {\top } }\otimes A)\operatorname {vec} (B)}
for matrices A, B, and C of dimensions k×l, l×m, and m×n. For example, if ad A ( X ) = A X − X A {\displaystyle \operatorname {ad} _{A}(X)=AX-XA} (the adjoint endomorphism of the Lie algebra gl(n, C) of all n×n matrices with complex entries), then vec ( ad A ( X ) ) = ( A ⊗ I n − I n ⊗ A ⊤ ) vec ( X ) {\displaystyle \operatorname {vec} (\operatorname {ad} _{A}(X))=(A\otimes I_{n}-I_{n}\otimes A^{\mathrm {\top } }){\text{vec}}(X)} , where I n {\displaystyle I_{n}} is the n×n identity matrix. There are two other useful formulations:
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