What is the inverse of the $\mbox{vec}$ operator?

Question

There is a well known vectorization operator $\mbox{vec}$ in matrix analysis.

I've vectorized my matrix equations, did some transformation of vectorized equations and now I want to get back to the matrix form. Is there special operator for it?

Answer 1

The inverse of the vectorization operator

$$\mbox{vec} : \mathbb{R}^{m \times n} \to \mathbb{R}^{mn}$$

is the operator

$$\mbox{vec}^{-1} : \mathbb{R}^{mn} \to \mathbb{R}^{m \times n}$$

such that

• $\mbox{vec}^{-1} (\mbox{vec} (X)) = X$ for all $X \in \mathbb{R}^{m \times n}$.

• $\mbox{vec} (\mbox{vec}^{-1} (x)) = x$ for all $x \in \mathbb{R}^{m n}$.

Once a matrix is vectorized, the original dimensions of the matrix are "forgotten". Hence, it would be wise to write the dimensions of the inputs of $\mbox{vec}$ and of the outputs of $\mbox{vec}^{-1}$ in subscripts, e.g., $\mbox{vec}_{m,n}$ and $\mbox{vec}_{m,n}^{-1}$. Note that $\mbox{vec}_{3,2}^{-1}$ is the inverse of $\mbox{vec}_{3,2}$, but not of $\mbox{vec}_{2,3}$.