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Matrix Vectorization

Matrix vectorization maps an m×n matrix A=(a_(ij)) to the mn×1 column vector formed by stacking its columns,

vec(A)=[a_(11) a_(21) ... a_(m1) a_(12) ... a_(m2) ... a_(1n) ... a_(mn)]^T.

For conformable matrices, vectorization satisfies

vec(AXB)=(B^T tensor A)vec(X),

where ^T denotes the transpose and tensor is the Kronecker product. This identity converts a Sylvester equation into a linear system of equations for the entries of its unknown matrix.

For an n×n symmetric matrix, half-vectorization keeps only the n(n+1)/2 entries on and below the main diagonal, avoiding duplication of the off-diagonal entries.

See also

Column Vector, Half-Vectorization, Kronecker Product, Lower Triangular Matrix, Matrix, Matrix Multiplication, Symmetric Matrix, Transpose

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References

Henderson, H. V. and Searle, S. R. "Vec and Vech Operators for Matrices, with Some Uses in Jacobians and Multivariate Statistics." Canad. J. Statist. 7, 65-81, 1979. https://doi.org/10.2307/3315017.Henderson, H. V. and Searle, S. R. "The Vec-Permutation Matrix, the Vec Operator and Kronecker Products: A Review." Linear Multilinear Algebra 9, 271-288, 1981. https://doi.org/10.1080/03081088108817379.Magnus, J. R. and Neudecker, H. Matrix Differential Calculus with Applications in Statistics and Econometrics, 3rd ed. Hoboken, NJ: Wiley, 2019. https://doi.org/10.1002/9781119541219.

Cite this as:

Weisstein, Eric W. "Matrix Vectorization." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MatrixVectorization.html

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