When working with matrices in Julia, it is often necessary to rank the elements within each column. This can be done in several ways, each with its own advantages and disadvantages. In this article, we will explore three different approaches to solving this problem.

## Approach 1: Using the `sortperm` function

One way to rank the elements in each column of a matrix is to use the `sortperm` function. This function returns the indices that would sort an array in ascending order. By applying this function to each column of the matrix, we can obtain the ranking of the elements.

```
function rank_columns(matrix)
num_cols = size(matrix, 2)
rankings = Array{Int}(undef, size(matrix))
for col in 1:num_cols
sorted_indices = sortperm(matrix[:, col])
rankings[:, col] = sorted_indices
end
return rankings
end
```

In this approach, we iterate over each column of the matrix and use the `sortperm` function to obtain the sorted indices. We then assign these indices to the corresponding column in the `rankings` array. Finally, we return the `rankings` array.

## Approach 2: Using the `sort!` function

Another way to rank the elements in each column of a matrix is to use the `sort!` function. This function sorts an array in place, meaning that it modifies the original array. By applying this function to each column of the matrix, we can obtain the ranking of the elements.

```
function rank_columns(matrix)
num_cols = size(matrix, 2)
rankings = Array{Int}(undef, size(matrix))
for col in 1:num_cols
sort!(matrix[:, col])
rankings[:, col] = 1:size(matrix, 1)
end
return rankings
end
```

In this approach, we iterate over each column of the matrix and use the `sort!` function to sort the elements in place. We then assign the indices from 1 to the number of rows in the `rankings` array. Finally, we return the `rankings` array.

## Approach 3: Using the `sortrows` function

A third way to rank the elements in each column of a matrix is to use the `sortrows` function. This function sorts the rows of a matrix based on the values in a specified column. By applying this function to each column of the matrix, we can obtain the ranking of the elements.

```
function rank_columns(matrix)
num_cols = size(matrix, 2)
rankings = Array{Int}(undef, size(matrix))
for col in 1:num_cols
sorted_matrix = sortrows(matrix, col)
rankings[:, col] = 1:size(matrix, 1)
end
return rankings
end
```

In this approach, we iterate over each column of the matrix and use the `sortrows` function to sort the rows based on the values in the column. We then assign the indices from 1 to the number of rows in the `rankings` array. Finally, we return the `rankings` array.

After comparing these three approaches, it is clear that Approach 1 using the `sortperm` function is the most efficient and concise solution. It avoids modifying the original matrix and provides the rankings directly. Therefore, Approach 1 is the recommended option for ranking over each matrix column sort in Julia.