The Hadamard Product (also known as the element-wise or Schur product) is an operation on two matrices of the same dimensions. The resulting matrix, C, is found by multiplying the corresponding elements of the input matrices, A and B.

Formula:

For two matrices A and B of size \(m \times n\), the Hadamard product \(C = A \circ B\) is a matrix of the same size where each element is defined as:

\(C_{ij} = A_{ij} \times B_{ij}\)

Example:

\( \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \circ \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 1 \times 5 & 2 \times 6 \\ 3 \times 7 & 4 \times 8 \end{pmatrix} = \begin{pmatrix} 5 & 12 \\ 21 & 32 \end{pmatrix} \)