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 by vec(A), is the mn × 1 column vector obtained by stacking the columns of the matrix A on top of one another:
For example, for the 2×2 matrix = , the vectorization is .
Compatibility with Kronecker products
for matrices A, B, and C of dimensions k×l, l×m, and m×n. For example, if (the adjoint endomorphism of the Lie algebra gl(n,C) of all n×n matrices with complex entries), then , where is the n×n identity matrix.
There are two other useful formulations:
Compatibility with Hadamard products
- vec(A B) = vec(A) vec(B).
Compatibility with inner products
- tr(A* B) = vec(A)* vec(B)
where the superscript * denotes the conjugate transpose.
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 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:
- vech(A) = [ A1,1, ..., An,1, A2,2, ..., An,2, ..., An-1,n-1,An-1,n, An,n ]T.
For example, for the 2×2 matrix A = , the half-vectorization is vech(A) = .
Programming languages that implement matrices may have easy means for vectorization. In Matlab/GNU Octave a matrix
A can be vectorized by
A(:). In Python NumPy arrays implement the 'flatten' method (although this stacks the rows of the matrix, not the columns), while in R the desired effect can be achieved via the 'c()' or 'as.vector()' functions.