Hadamard Calculator

Hadamard Product:

\[ A \circ B = \begin{bmatrix} a_{11} \times b_{11} & a_{12} \times b_{12} & \cdots \\ a_{21} \times b_{21} & a_{22} \times b_{22} & \cdots \\ \vdots & \vdots & \ddots \end{bmatrix} \]

Hadamard Product (A ∘ B):

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1. What is Hadamard Product?

The Hadamard product (also known as the element-wise product) is a binary operation that takes two matrices of the same dimensions and produces another matrix where each element is the product of the corresponding elements from the original matrices.

2. How Does the Calculator Work?

The calculator computes the Hadamard product using the formula:

\[ (A \circ B)_{ij} = A_{ij} \times B_{ij} \]

Where:

\( A \) — First input matrix
\( B \) — Second input matrix (must be same size as A)
\( \circ \) — Hadamard product operator
\( i,j \) — Row and column indices

Explanation: Each element in the resulting matrix is simply the product of the corresponding elements in the input matrices.

3. Importance of Hadamard Product

Details: The Hadamard product is widely used in machine learning, image processing, and neural networks. It's particularly important in convolutional neural networks (CNNs) and various filtering operations.

4. Using the Calculator

Tips: Enter matrices with elements separated by spaces and rows separated by newlines. Both matrices must have exactly the same dimensions. Decimal numbers are supported.

5. Frequently Asked Questions (FAQ)

Q1: How is Hadamard product different from matrix multiplication?
A: Hadamard product is element-wise while matrix multiplication involves dot products of rows and columns.

Q2: Can I multiply matrices of different sizes?
A: No, for Hadamard product both matrices must have identical dimensions.

Q3: What are some applications of Hadamard product?
A: Image filtering, masking operations, attention mechanisms in neural networks, and various pointwise transformations.

Q4: Does the calculator support complex numbers?
A: This version only supports real numbers. Complex number support may be added in future versions.

Q5: What's the computational complexity of Hadamard product?
A: O(n²) for n×n matrices, as it requires computing each element individually.