Matrix Multiplications

Let \(\mathbf{A} \in \mathbb{C}^{n \times m}, \mathbf{X} \in \mathbbf{X}^{m\times k}\). The matrix product \(\mathbf{B}=\mathbf{AX}\) is defined to be the matrix such that \(\mathbf{B}_{ij}=\sum_{k=1}^m \mathbf{A}_{ik}\mathbf{X}_{kj}\).

This can also be viewed as each column \(\mathbf{b}_{:,j} \in \mathbb{R}^k\) being a linear combination of each of the \(m\) columns of \(\mathbf{A}\) with the coefficients being the elements in row \(\mathbf{x}_{j,:} \in \mathbb{R}^m\).

That is, \(\mathbf{b}_{:,j} = \sum_{l=1}^k \mathbf{x}_{jk}\mathbf{A}_{:,l}\).

In the special case that \(k=1\) i.e. \(\mathbf{b}\) is a vector, the matrix-vector product \(\mathbf{Ax}=\mathbf{b}\) is originally defined to be the vector such that \(\mathbf{b}_i = \sum_{k=1}^m \mathbf{A}_{ik}\mathbf{x}_k\).

Similarly to before, this could be written as taking a linear combinations of the columns of \(\mathbf{A}\) with the coefficients beings the elements in \(\mathbf{x}\).

That is, \(\mathbf{b} = \sum_{k=1}^m \mathbf{x}_k\mathbf{A}_{:,k}\).

The original definitions of \(\mathbf{Ax}=\mathbf{b}\) can be viewed as the matrix \(\mathbf{A}\) acting on \(\mathbf{x}\) to produce some result, meanwhile the definition in sums in of column vectors can be thought of as \(\mathbf{x}\) acting on columns of \(\mathbf{A}\) to produce some result.