There are great answers by fellow members. I would like to visualize just this particular problem. Lets say there are $4$ companies $A$,$B$,$C$ and $D$ and all of them sell three fruits Apples, Oranges and Pears. Because the numbers are less, I will assume that we want to see the daily sales in numbers of all companies.
Create the table for daily sales:
$$\begin{bmatrix} &\text {Apples} & \text{Oranges}&\text{Pears} \\\text{Company 1}&10&2&5\\\text{Company 2} &5&3&10\\\text{Company 3} &4&3&2\\\text{Company 4} &5&10&5\\\end{bmatrix}$$
Just ignore the words and look at the numbers. The first row and first column are just for understanding. The numerical values of the table represent your matrix $A$. This table tells you the daily sales of each company for apples, oranges and pears.
$$A=\begin{bmatrix}10 & 2&5 \\5 & 3&10 \\4 & 3&2\\5 & 10&5 \end{bmatrix}$$
If we just write the table in another way, to see just the sales of a particular fruit from all the companies we will write,
$$\begin{bmatrix} &\text {Company 1} & \text{Company 2}&\text{Company 3}&\text{Company 4} \\\text{Apples}&10&5&4&5\\\text{Oranges} &2&3&3&10\\\text{Pears} &5&10&2&5\\\end{bmatrix}$$
This can be written as:
$$A^T=\begin{bmatrix}10 & 5&4&5 \\2 &3& 3&10 \\5 & 10&2&5\\ \end{bmatrix}$$
Now we keep both the tables together,
$$\begin{bmatrix} &\text {Apples} & \text{Oranges}&\text{Pears} \\\text{Company 1}&10&2&5\\\text{Company 2} &5&3&10\\\text{Company 3} &4&3&2\\\text{Company 4} &5&10&5\\\end{bmatrix}\begin{bmatrix} &\text {Company 1} & \text{Company 2}&\text{Company 3}&\text{Company 4} \\\text{Apples}&10&5&4&5\\\text{Oranges} &2&3&3&10\\\text{Pears} &5&10&2&5\\\end{bmatrix}$$
If by some case there is a partnership between two companies say Company A and Company B, then what will be the total fruit sales?
$$\text{Total fruit sales for the partnership} = \text{No of total apples + No of total oranges + No of total pears}$$
Total fruit sales for the partnership = Company 1 Apples X Company 2 Apples + Company 1 Oranges X Company 2 Oranges + Company 1 Pears X Company 2 Pears
$$\text{Total fruit sales for the partnership} = 10X5 + 2X3 + 5X10=106$$
So the total sales of fruits for the partnership of Company A and Company B is $106$.
This is nothing but the second element of the product $AA^T$.
$$AA^T=\begin{bmatrix}10 & 2&5 \\5 & 3&10 \\4 & 3&2\\5 & 10&5 \end{bmatrix}\begin{bmatrix}10 & 5&4&5 \\2 &3& 3&10 \\5 & 10&2&5\\ \end{bmatrix}=\begin{bmatrix}129&106&56&85 \\106&134&49&105 \\56&49&29&60\\ 85&105&60&150\end{bmatrix}$$
What does this product show? This product can be visualized as the total sales chart of each company as well as the total sales of mutual parnterships of companies.
$$\begin{bmatrix} &\text {Company 1} & \text{Company 2}&\text{Company 3}&\text{Company 4} \\\text{Company 1}&129&106&56&85\\\text{Company 2} &106&134&49&105\\\text{Company 3} &56&49&29&60\\\text{Company 4} &85&105&60&150\\\end{bmatrix}$$
Crucial points to observe:
The diagonal elements of the matrix $AA^T$ are all just the squared sum of individual companies. For example the first element is the strength of sales of Company 1 and so on.
Each non diagonal element shows the total sales that would result due to the partnership between two companies. For example the second element of $AA^T$ is the total sales produced due to the partnership between Company 1 and Company 2.
The matrix $AA^T$ is symmetric, which can be visualized using the fact that the total sales due to the partnership of Company 1 and Company 2 is same as that of Company 2 and Company 1.
Useful insight from $AA^T$is that check the diagonal elements , whichever is the maximum, you can confirm that Company is stronger in sales. Another useful insight is you can check whether partnership with a particular company is beneficial or not. For example, Company 3 is having the lowest sales individually, so it is beneficial for Company 3 to form a partnership with Company 4 because the total sales would be 60 which is more than double of what Company 3 can have. So, we can check which partnerships would be most beneficial.
Diagonal elements: (A measure of) Individual strengths, Non Diagonal Elements: Partnership strengths.
Hope this helps...
Best Answer
These kind of matrices are called orthogonal matrices . There are many examples of them, notably the following rotation matrix:
\begin{bmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \\ \end{bmatrix} Note that this makes use that the determinant $cos^2(\theta) + sin^2(\theta)$ will always be one. Furthermore, the adjoint of this matrix will equal the transpose. Another example would be the identity matrix:
\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}
I would like to note that these kind of matrices have many interesting properties. Notably, the set of all the rows or the set of all the columns will be a set of orthonormal vectors.
A 2x2 matrix will be orthogonal if the following criteria are met:
\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}
$$ 1 = a^2 + x^2 $$ $$ 1 = b^2 + c^2 $$ $$ 0 = ac + bd $$
There are indeed strategies of finding orthogonal matrices of higher dimensions.