Understanding Premultiplication in Linear Algebra

Premultiplication is a matrix operation that multiplies each element of one matrix by the corresponding element of another matrix, and is denoted by the symbol "·" or "⋅". It is also known as the Hadamard product or the Schur product.

In more detail, if we have two matrices A and B, their premultiplication AB is defined as follows:

(AB)ij = ∑k=1n AkijBkj

where A is an n x n matrix, B is an n x m matrix, and n and m are the dimensions of the matrices. The result is an n x m matrix, where each element at position (i, j) is the sum of the products of the corresponding elements of A and B.

Premultiplication has some useful properties, such as:

* (AB)B = A(BB) = A(A^T) = AA^T
* (AB)^T = B^T A^T = (BA)^T
* (AB) + (AC) = (A+C)B
* (AB) - (AC) = A(B-C)

Premultiplication is used in many areas of linear algebra, such as eigenvalue decomposition, singular value decomposition, and matrix factorization. It is also used in machine learning, signal processing, and other fields where matrices are used to represent data or transformations.