How to convert an array of array into a matrix?

As you noted, you can concatenate an array of arrays x using hcat(x...), but it's usually better to create a matrix to begin with instead. Two ways that you can do it in this case:

1. Using broadcasting:

   phi(x, d) = x.^((0:d)')
   

   As long as x is a vector, it will broadcast against the row matrix (0:d)'.

   You can get the transpose result by transposing x instead of the range 0:d.

2. Using a two-dimensional array comprehension:

   phi(x, d) = [xi.^di for xi in x, di in 0:d]
   

   This will work as long as x is iterable. If x is an n-d array, it will be interpreted as if it were flattened first.

   You can transpose the result by switching the order of the comprehension variables:

   phi(x, d) = [xi.^di for di in 0:d, xi in x]
   

Converting phi's output to a matrix can be done as follows:

   y = hcat(phi(x, 3)...)

or if you prefer the vectors to be rows, a transpose is needed:

   y = vcat([x' for x in phi(x, 3)]...)

Alternatively, you can convert to a matrix within phi by defining it:

   phi(x, d) = hcat([x.^i for i=0:d]...)

More generally you can also use the splat operator ... with hcat:

X = [[1,2], [3, 4], [5,6]]
hcat(X...)

gives a 2-by-3 matrix

1  3  5
2  4  6

An alternative to splatting like hcat(X...) is to use reduce

reduce(hcat, X)

So, in your case, this will do the trick

reduce(hcat, phi(x, 3))