You need to answer the question chiro asked, first. What kind of "multiplication" are you using? You have to have a square before you can talk about a square root! There are two different ways to multiply a vector by a vector in three dimensions: the dot product of a vector with itself gives a number (the square of the length of the vector) not a vector so there is no square root in that sense. The cross product of a vector with itself is always the 0 vector so there is no square root of a vector in that sense.
You talk about writing the vector as a column matrix so perhaps you are talking about matrix multiplication. But you multiply an "m by n" matrix by an "n by p" matrix- the number of columns of the first matrix has to be the same as the same as the number of rows of the second matrix and that is not possible for an "n by 1" matrix multiplied by itself. Some texts define the square of a column matrix as the product of the matrix with its transpose- but that gives a number (strictly speaking a "one by one matrix"), so that is equivalent to the dot product, and there is, again, no "square root" of a vector.
That is why we do NOT in general define the "square root of a vector". If you have been given a problem that asks you to find the square root of a vector, you are going to have to tell us what definition of "square root of a vector" you are using.