Applying the vec operator

Question

How to apply the $\operatorname{vec}$ operator in Mathematica? For example, how can I transform a $2 \times 2$ matrix into a $1 \times 4$ matrix as follows? $$ \operatorname{vec}\left( \begin{bmatrix} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \end{bmatrix} \right) = \begin{bmatrix} a_{1,1} \\ a_{2,1} \\ a_{1,2} \\ a_{2,2} \end{bmatrix} $$

Answer 1

Use Flatten to do this in a single operation.

in = Array[a, {2, 2}]

Flatten[in, {2, 1}]

{{a[1, 1], a[1, 2]}, {a[2, 1], a[2, 2]}}
{a[1, 1], a[2, 1], a[1, 2], a[2, 2]}

In Mathematica there are only vectors (lists), not column vectors and row vectors. However if you wish to convert a vector into an array with rows of length one for output the computationally fastest method is typically Partition:

Partition[{a[1, 1], a[2, 1], a[1, 2], a[2, 2]}, 1]

{{a[1, 1]}, {a[2, 1]}, {a[1, 2]}, {a[2, 2]}}

If that is your goal from the start you can also do that in a single operation using Flatten:

Flatten[{in}, {3, 2}]     (* note the extra {} around in *)

{{a[1, 1]}, {a[2, 1]}, {a[1, 2]}, {a[2, 2]}}

Recommended reading:

• Flatten command: matrix as second argument

Answer 2

{{1, 2}, {4, 5}} // MatrixForm

\begin{bmatrix} 1 & 2 \\ 4 & 5 \end{bmatrix}

ArrayReshape[Transpose[%], {4, 1}] // MatrixForm

\begin{bmatrix} 1 \\ 4\\ 2\\ 5 \end{bmatrix}

Thanks to @cyrille.piatecki for the use of Transpose[].

Answer 3

I propose this simple module

vec[mat_] := 
 Module[{a = mat}, 
  ArrayReshape[Transpose[
   a], {Dimensions[mat][[1]] Dimensions[mat][[2]], 1}]]

apply with

aa = Table[Subscript[a, i, j], {i, 1, 2}, {j, 1, 2}]

this gives the expected result

vec[aa] // MatrixForm