# ECSE 506: Stochastic Control and Decision Theory

Vectorization

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