- Vectorization (mathematics)
In
mathematics , especially inlinear algebra andmatrix theory , the vectorization of a matrix is alinear transformation which converts the matrix into acolumn vector . Specifically, the vectorization of an "m×n" matrix "A", denoted by vec("A"), is the "mn × 1" column vector obtain by stacking the columns of the matrix "A" on top of one another::Here represents the -th element of matrix and the superscript denotes the
transpose .For example, for the 2×2 matrix = , the vectorization is .
Compatibility with Kronecker products
The vectorization is frequently used together with the
Kronecker product to expressmatrix multiplication as a linear transformation on matrices. In particular,:for matrices "A", "B", and "C" of compatible dimensions. For example, if ad"A"("X") = "AX" − "XA" (the
adjoint endomorphism of theLie algebra gl("n",C) of all "n×n" matrices with complex entries), then vec(ad"A"("X")) = ("I" "A" − "A"T "I") vec("X"), where"I" is the "n×n"identity matrix .There are two others useful formulations:
:
:
Compatibility with Hadamard products
Vectorization is an
algebra homomorphism from the space of "n×n" matrices with the Hadamard (entrywise) product to Cn with its Hadamard product::vec("A" "B") = vec("A") vec("B").
Compatibility with inner products
Vectorization is a
unitary transformation from the space of "n×n" matrices with the Frobenius (or Hilbert-Schmidt)inner product to Cn ::tr("A"* "B") = vec("A")* vec("B")
where the superscript * denotes the
conjugate transpose .Half-vectorization
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 themain 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") = [ "A"1,1, ..., "A"m,1, "A"2,2, ..., "A"n,2, ..., "A"n-1,n-1,"A"n-1,n, "A"n,n ] T.For example, for the 2×2 matrix "A" = , the half-vectorization is vech("A") = .
There exist unique matrices transforming the half-vectorization of a matrix to its vectorization and vice-versa called, respectively, the
duplication matrix and theelimination matrix .ee also
*
Voigt notation
* Column-major orderReferences
*Jan R. Magnus and Heinz Neudecker (1999), "Matrix Differential Calculus with Applications in Statistics and Econometrics", 2nd Ed., Wiley. ISBN 0-471-98633-X.
*Jan R. Magnus (1988), "Linear Structures", Oxford University Press. ISBN 0-85264-299-7.
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