Vectorization (mathematics)

In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a column vector. Specifically, the vectorization of an 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: In mathematics, especially in linear algebra and matrix theory, the vectorization of a matrix is a linear transformation which converts the matrix into a column vector. Specifically, the vectorization of an 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: Here, a i , j {displaystyle a_{i,j}} represents A ( i , j ) {displaystyle A(i,j)} and the superscript T {displaystyle {}^{mathrm {T} }} denotes the transpose. Vectorization expresses, through coordinates, the isomorphism R m × n := R m ⊗ R n ≅ R m n {displaystyle mathbf {R} ^{m imes 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 A {displaystyle A} = [ a b c d ] {displaystyle {egin{bmatrix}a&b\c&dend{bmatrix}}} , the vectorization is v e c ( A ) = [ a c b d ] {displaystyle mathrm {vec} (A)={egin{bmatrix}a\c\b\dend{bmatrix}}} . The vectorization is frequently used together with the Kronecker product to express matrix multiplication as a linear transformation on matrices. In particular, 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 { ext{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 ) ) = ( I n ⊗ A − A T ⊗ I n ) vec ( X ) {displaystyle { ext{vec}}({ ext{ad}}_{A}(X))=(I_{n}otimes A-A^{mathrm {T} }otimes I_{n}){ ext{vec}}(X)} , where I n {displaystyle I_{n}} is the n×n identity matrix.