ECSE 506: Stochastic Control and Decision Theory

Aditya Mahajan
Winter 2022

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Vectorization is a linear transformation that converts a matrix to a column vector. For example, \[\VEC\left(\MATRIX{a & b \\ c & d }\right) = \MATRIX{a \\ c \\ b \\ d}.\]

Vectorization is often used to express matrix multiplication as a linear transformation on matrices. In particular, we have the following three properties:

  1. \(\VEC(ABC) = (C^\TRANS \otimes A) \VEC(B).\)
  2. \(\VEC(ABC) = (I \otimes AB)\VEC(C).\)
  3. \(\VEC(ABC) = (C^\TRANS B^\TRANS \otimes I)\VEC(A).\)

Another useful formulation is the following

  1. \(\VEC(AB) = (I \otimes A) \VEC(B) = (B^\TRANS \otimes I) \VEC(A).\)

Vectorization is also useful to view trace as an inner product. In particular,

\[\TR(A^\TRANS B) = \VEC(A)^\TRANS \VEC(B)\]

This entry was last updated on 12 Oct 2019