Transpose Array

1. Background Knowledge

NumPy arrays are the foundational data structure for numerical computing in Python, enabling efficient vectorized operations on multi-dimensional data. A 2D array (matrix) of shape (m, n) has m rows and n columns. The transpose operation swaps these dimensions, converting shape (m, n) to (n, m) where element at position (i, j) moves to (j, i).

Mathematically, for matrix $A$, the transpose $A^T$ satisfies: $(A^T)_{j,i} = A_{i,j}$

NumPy provides two primary methods: .T attribute (simple view, no copy for 2D) and np.transpose() (more flexible for >2D).

Prerequisites:

• Basic NumPy: np.array(), .shape
• Array views vs. copies
• Dictionary manipulation in Python

2. Algorithm Approach

Direct transposition using NumPy's optimized C-level implementation:

1. Access array shape: arr.shape
2. Compute transpose: arr.T or np.transpose(arr)
3. Convert to nested list: .tolist() (required by output format)

NumPy's transpose creates a view (zero-copy for 2D arrays), making it $O(1)$ time and space for the operation itself.

Why not manual loops? NumPy's vectorization is 100-1000x faster than Python loops due to C implementation and cache optimization.

3. Step-by-Step Strategy

def transpose_array(arr):
 # Step 1: Capture original shape as list
 original_shape = list(arr.shape)

 # Step 2: Transpose using.T (or np.transpose(arr))
 transposed = arr.T

 # Step 3: Convert to nested list (required format)
 transposed_list = transposed.tolist()

 # Step 4: Capture new shape
 transposed_shape = list(transposed.shape)

 # Step 5: Return dictionary
 return {
 "original_shape": original_shape,
 "transposed_shape": transposed_shape,
 "array": transposed_list
 }

Verification with sample:

Input: [[1,2,3], [4,5,6]] # shape (2,3)
Output: [[1,4], [2,5], [3,6]] # shape (3,2)

4. Common Pitfalls

• Returning NumPy array instead of nested list: Use .tolist()—problem explicitly requires "array": nested list.
• Shape as tuple: Convert with list(arr.shape)—requires lists.
• Using .transpose() method: Correct is .T attribute or np.transpose().
• Modifying original: .T creates view; use arr.copy().T if mutation avoidance needed (not required here).
• Non-2D input: Constraints guarantee 2D, but .T handles higher-D safely.
• Order of keys: Dictionary order preserved in Python 3.7+ matches requirement.

5. Time & Space Complexity

Time Complexity: $O(mn)$

• .T operation: $O(1)$ (creates metadata view)
• .tolist(): $O(mn)$ (copies all elements to Python lists)
• Shape operations: $O(1)$

Space Complexity: $O(mn)$

• Output nested list: stores full copy of transposed data
• Dictionary overhead: $O(1)$
• NumPy view: $O(1)$ extra (not counted in final output)

Total: $O(mn)$ time and space, where $m, n$ are original dimensions. Optimal—no better asymptotic possible for full materialization.

Edge Cases Handled:

• Square matrices: (n,n) → (n,n)
• 1-row/1-col: (1,n) → (n,1) ✓
• Empty: Constraints imply non-empty 2D.