1.1 Basic algorithms and notations

The matrix calculation is based on the linear operation. The point multiplication involves the addition and multiplication of scalar operations. Matrix-vector multiplication is made up of point multiplication. Matrix-matrix multiplication can be attributed to the collection of matrix-vector multiplication. All of these operations can be described in the form of algorithms or in the language of linear algebra. One of our goals is to show how the descriptions of these two styles complement each other. In this process, we will explain the notation and familiarize the reader with the way of thinking that supports the field of matrix computing. These discussions revolve around matrix multiplication, which can be organized in a number of different ways.

1.1.1 Matrix Symbol

Make the $\mathbb{r}$ represent the real number collection. We use $\mathbb{r}^{m\times n}$ to represent the vector space of all the m-by-n real matrices:

$A \in \mathbb{r}^{m\times n}\leftrightarrow a=\left (A_{ij} \right) =\begin{bmatrix}
A_{11} & \cdots & a_{1n} \ \
\vdots & & \vdots \ \
A_{M1} & \cdots & A_{mn}
\end{bmatrix},a_{ij}\in \mathbb{r}$

If the uppercase letter is used to refer to a matrix (for example, $A, B,\delta $), then the lowercase letter corresponding to the subscript $ij$ refers to the $\left (i,j \right) $ element (for example, $a _{ij},b_{ij},\delta _{ij}$).

1.1 Basic algorithms and notations