matrix standardization, inferring technique

Question

I am reading a research paper in which authors perform a matrix standardization but do not explain the actual procedure. The original 3x3 matrix is:

411; 124; 45
112; 17;  50
 20; 378; 285

the standardized is:

0;      0.7036;   1
0.7647;   1;      0.9792
1;        0;      0

the minimum of each column (and row) becomes 1, the maximum of each column becomes 0, but I don't get how the other values are calculated. Any idea? Thank you

Answer 1

Let $\textbf{a}_j=\{a_{1j},\ldots, a_{ij}, \ldots, a_{mj}\}$. Then the formula is

$$\tilde a_{ij}=\frac{\max\limits_i(\textbf{a}_j)-a_{ij}}{\max\limits_i(\textbf{a}_j)-\min\limits_i(\textbf{a}_j)}$$

$i$ is the index for the row and $j$ is the index for the column. For example,

$$\tilde a_{13}=\frac{\max\{45,50,285\}-45}{\max\{45,50,285\}-\min\{45,50,285\}}=\frac{285-45}{285-45}=1$$