A Tensor is a container which can house data in N dimensions. Tensors are in fact generalizations of matrices to N-dimensional spaces, which is why they’re often used interchangeably with the matrix, (which is specifically a 2-dimensional or Rank2 tensor).\(^1\)

  1. Tensor product for the vector:components of the first vector are multiplied by the second vector element wise. \[ \begin{bmatrix} a_1\\ a_2 \end{bmatrix} \bigotimes \begin{bmatrix} b_1\\ b_2\\ b_3 \end{bmatrix}=\begin{bmatrix} a_1(\begin{bmatrix} b_1\\ b_2\\ b_3 \end{bmatrix})\\ a_2(\begin{bmatrix} b_1\\ b_2\\ b_3 \end{bmatrix}) \end{bmatrix}=\begin{bmatrix} a_1b_1\\ a_1b_2\\ a_1b_3\\ a_2b_1\\ a_2b_2\\ a_2b_3 \end{bmatrix} \]
  2. Tensor product for matrix:

Let \(X=\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}, I=\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\), then

\[\begin{align*} X\bigotimes I&=\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}\bigotimes \begin{bmatrix} 1 & 0\\ 0 & 1\end{bmatrix}\\&=\begin{bmatrix} 0(I) & 1(I)\\ 1(I) & 0(I) \end{bmatrix}\\&=\begin{bmatrix} \begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix}& \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\\ \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} & \begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix} \end{bmatrix} \end{align*}\]

Reference

1.https://medium.com/quantum-untangled/tensor-products-linear-algebra-for-qc-8f7bb5020c6c