Hadamard Product Calculator
Calculate the element-wise product of two matrices with step-by-step explanations
Matrix Dimensions
Matrix A
Matrix B
Hadamard Product Result
Enter values in the matrices to see the Hadamard product
The Hadamard product multiplies corresponding elements: (A ∘ B)[i,j] = A[i,j] × B[i,j]
Example: 2×2 Hadamard Product
Input Matrices
Matrix A:
[1, 4]
Matrix B:
[3, 1]
Hadamard Product A ∘ B
Element (1,1): 2 × 5 = 10
Element (1,2): 3 × 2 = 6
Element (2,1): 1 × 3 = 3
Element (2,2): 4 × 1 = 4
[3, 4]
Hadamard Product Properties
Element-wise
(A ∘ B)[i,j] = A[i,j] × B[i,j]
Multiply corresponding elements
Commutative
A ∘ B = B ∘ A
Order doesn't matter
Same Dimensions
Matrices must have identical size
Result has same dimensions
Key Properties
Hadamard Product
(A ∘ B)[i,j] = A[i,j] × B[i,j]
Commutativity
A ∘ B = B ∘ A
Associativity
A ∘ (B ∘ C) = (A ∘ B) ∘ C
Distributivity
A ∘ (B + C) = A ∘ B + A ∘ C
Understanding the Hadamard Product
What is the Hadamard Product?
The Hadamard product, also known as the element-wise product or Schur product, is a binary operation that takes two matrices of the same dimensions and produces another matrix by multiplying corresponding entries. It's denoted by the symbol ∘ (small circle).
Key Characteristics
- •Element-wise operation: Each element is multiplied independently
- •Same dimensions required: Both input matrices must be identical in size
- •Result size: Output matrix has the same dimensions as inputs
- •Commutative: A ∘ B = B ∘ A (unlike standard matrix multiplication)
Applications
- •Image processing and computer vision
- •Neural networks and deep learning
- •Signal processing and filtering
- •Statistics and data analysis
- •Quantum computing and tensor operations
Note: The Hadamard product is different from standard matrix multiplication. While matrix multiplication combines rows and columns, the Hadamard product simply multiplies corresponding elements.
Mathematical Properties
Commutativity:
A ∘ B = B ∘ A
Associativity:
A ∘ (B ∘ C) = (A ∘ B) ∘ C
Distributivity over addition:
A ∘ (B + C) = (A ∘ B) + (A ∘ C)
Identity element:
A ∘ J = A, where J is a matrix of all ones
Rank inequality:
rank(A ∘ B) ≤ rank(A) × rank(B)