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 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: Here, represents the element in the i-th row and j-th column of A, and the superscript denotes the transpose. Vectorization expresses, through coordinates, the isomorphism between these (i.e., of matrices and vectors) as vector spaces. For example, for the 2×2 matrix , the vectorization is . The connection between the vectorization of A and the vectorization of its transpose is given by the commutation matrix. 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 (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: More generally, it has been shown that vectorization is a self-adjunction in the monoidal closed structure of any category of matrices. Vectorization is an algebra homomorphism from the space of n × n matrices with the Hadamard (entrywise) product to Cn2 with its Hadamard product: Vectorization is a unitary transformation from the space of n×n matrices with the Frobenius (or Hilbert–Schmidt) inner product to Cn2: where the superscript † denotes the conjugate transpose. The matrix vectorization operation can be written in terms of a linear sum. Let X be an m × n matrix that we want to vectorize, and let ei be the i-th canonical basis vector for the n-dimensional space, that is . Let Bi be a (mn) × m block matrix defined as follows: Bi consists of n block matrices of size m × m, stacked column-wise, and all these matrices are all-zero except for the i-th one, which is a m × m identity matrix Im.