Vectorization is a transformation of a matrix into a vector, usually by stacking its columns into a column vector. For example, in Matlab, matrix A can be converted to a column vector by A(:). This is a linear transformation. I guess this means that it can be written as a multiplication of matrix A by one or more matrices, but how?
Thanks everyone. Both Matlab and C/C++ can do efficient vectorization via linear indexing. My question was not about implementation, but about how to represent vectorization as a linear transformation - this comes in handy if you want to derive or prove results that involve vectorization.
Anyway, with the help of your input, I found the answer. While it is not possible to express vectorization as a single matrix multiplication of the initial matrix and another matrix, it is possible to express it as a sum of several terms, each involving matrix-vector multiplication. If interested, see attached document.
Use the canonical basis vectors e_1, e_2, ....
Columns 1,2,... of A is given by A e_1, A e_2, ....
Stack them onto one vector
Thanks very much, Sarod. I understand column extraction, but can stacking be represented as a matrix multiplication?
Hi,
I don't know if was exactly the question, but this procedure - transform a matrix into a vector - is quite common when you program an algorithm. In this case you simply address the elements by the following rule:
index_vector = index_col*N_line + index_line
where:
index_vector - vector index
index_col - matrix column index
index_line - matriz line index
N_line - matrix line number --> A[N_line,N_column]
best regards
Any linear transformation between R^M and R^N can be written in the form of a matrix multiplication. However, vectorization maps between R^MxN and R^P (where P = MxN), hence the property does not hold. See also:
http://en.wikipedia.org/wiki/Vectorization_(mathematics)
Thanks everyone. Both Matlab and C/C++ can do efficient vectorization via linear indexing. My question was not about implementation, but about how to represent vectorization as a linear transformation - this comes in handy if you want to derive or prove results that involve vectorization.
Anyway, with the help of your input, I found the answer. While it is not possible to express vectorization as a single matrix multiplication of the initial matrix and another matrix, it is possible to express it as a sum of several terms, each involving matrix-vector multiplication. If interested, see attached document.
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