can someone explain me the diffrence between tensor product of matrices and outer product of matrices?
The tensor product and outer product are both mathematical operations used in linear algebra to combine two vectors or matrices. The main difference between them is the type of output they produce and the mathematical properties that they follow.
The output of a tensor product is a tensor, which is a multidimensional array that represents the combined information of the two input vectors or matrices. The dimensions of the tensor are the product of the dimensions of the input vectors or matrices.
The output of an outer product is a matrix, which is a two-dimensional array that represents the outer product of the two input vectors or matrices. The dimensions of the matrix are the product of the dimensions of the input vectors or matrices.
The tensor product is commutative, meaning the order of the input vectors or matrices does not affect the output. It is also associative, meaning the grouping of the input vectors or matrices does not affect the output. Additionally, it follows the distributive property, meaning it distributes over addition and scalar multiplication.
The outer product is not commutative, meaning the order of the input vectors or matrices affects the output. It is also not associative, meaning the grouping of the input vectors or matrices affects the output. It does not follow the distributive property, as the outer product of two vectors cannot be expressed as the sum of the outer products of their individual components.