Matrix Operations

Matrix Operations

Matrices support element-wise operations and transpose.

A = [[1, 2], [3, 4]]
B = [[5, 6], [7, 8]]

# Element-wise addition
add = [[A[i][j] + B[i][j] for j in range(len(A[0]))] for i in range(len(A))]
print(add)    # [[6, 8], [10, 12]]

# Scalar multiplication
scaled = [[3 * A[i][j] for j in range(len(A[0]))] for i in range(len(A))]
print(scaled) # [[3, 6], [9, 12]]

Transpose

The transpose $mathbf{A}^T$ flips a matrix along its diagonal — rows become columns:

A = [[1, 2, 3],
     [4, 5, 6]]

AT = [[A[i][j] for i in range(len(A))] for j in range(len(A[0]))]
print(AT)
# [[1, 4], [2, 5], [3, 6]]
# rows: len(AT) = 3, cols: len(AT[0]) = 2

Symmetric Matrices

A matrix is symmetric if $mathbf{A} = mathbf{A}^T$. Covariance matrices and Gram matrices are always symmetric.

Your Task

Implement transpose(A) that returns the transpose of a matrix as a list of lists.