⊙ Hadamard Product Calculator
Calculate element-wise multiplication A ⊙ B
How to Use This Calculator
Select Matrix Dimensions
Choose rows and columns (both matrices must have same size).
Enter Matrices A and B
Input elements of both matrices.
Calculate
Click to compute element-wise product: (A ⊙ B)ᵢⱼ = Aᵢⱼ × Bᵢⱼ
Formula
(A ⊙ B)ᵢⱼ = Aᵢⱼ × Bᵢⱼ
Element-wise multiplication (each element multiplied separately)
Definition:
The Hadamard product (also called Schur product or element-wise product) multiplies corresponding elements: (A ⊙ B)ᵢⱼ = Aᵢⱼ × Bᵢⱼ
Requirements:
Matrices A and B must have the same dimensions (m × n)
Example:
A = [1 2; 3 4], B = [5 6; 7 8]
A ⊙ B = [1×5 2×6; 3×7 4×8] = [5 12; 21 32]
Properties:
- Commutative: A ⊙ B = B ⊙ A
- Associative: (A ⊙ B) ⊙ C = A ⊙ (B ⊙ C)
- Distributive: A ⊙ (B + C) = (A ⊙ B) + (A ⊙ C)
About Hadamard Product Calculator
The Hadamard Product Calculator computes the element-wise multiplication of two matrices. Unlike standard matrix multiplication, the Hadamard product multiplies corresponding elements position by position. It's denoted by ⊙ and is widely used in signal processing, neural networks, and optimization.
When to Use This Calculator
- Neural Networks: Element-wise operations in deep learning
- Signal Processing: Component-wise multiplication of signals
- Image Processing: Pixel-wise operations
- Optimization: Gradient updates in optimization algorithms
- Statistics: Variance calculations and element-wise operations
Why Use Our Calculator?
- ✅ Simple Operation: Element-wise multiplication
- ✅ Flexible Size: Supports various matrix dimensions
- ✅ Clear Display: Shows both input and result matrices
- ✅ Educational: Helps understand Hadamard product
- ✅ Accurate: Precise calculations
- ✅ Free: No registration required
Key Concepts
- Element-wise: Each element (A ⊙ B)ᵢⱼ = Aᵢⱼ × Bᵢⱼ
- Same Dimensions: A and B must be same size (unlike standard multiplication)
- Not Matrix Multiplication: Different from AB (dot product)
- Symbol: Denoted by ⊙ (circle with dot) or sometimes ∘
- Identity: Element-wise identity matrix has all 1s
Difference from Standard Multiplication
Standard (Dot Product): (AB)ᵢⱼ = Σₖ AᵢₖBₖⱼ (rows × columns, requires compatible dimensions)
Hadamard (Element-wise): (A ⊙ B)ᵢⱼ = Aᵢⱼ × Bᵢⱼ (position-by-position, requires same dimensions)
Frequently Asked Questions
What is Hadamard product?
Hadamard product (⊙) is element-wise multiplication: (A ⊙ B)ᵢⱼ = Aᵢⱼ × Bᵢⱼ. Each element is multiplied with the corresponding element in the same position.
How is Hadamard product different from matrix multiplication?
Matrix multiplication AB computes dot products (rows × columns) and requires compatible dimensions. Hadamard product A ⊙ B multiplies element-by-element and requires identical dimensions.
What are the requirements?
Both matrices must have exactly the same dimensions (same number of rows and columns). Unlike standard multiplication, you can't multiply 2×3 with 3×2 using Hadamard product.
Is Hadamard product commutative?
Yes! A ⊙ B = B ⊙ A because multiplication of numbers is commutative. Standard matrix multiplication is not commutative in general.
Where is Hadamard product used?
Commonly used in neural networks (element-wise operations in layers), signal processing (component-wise filtering), image processing (pixel operations), and optimization (gradient updates).