How does this matrix multiply work in SQL? - mysql

Question

Full disclosure, I'm a noob at SQL
Given two sparce matrices A and B, defined as:
A(row_number, column_number, value) and B(row_number, column_number, value)
I don't understand how this query represents the multiplication of the two matrices:
SELECT A.row_number, B.column_number, SUM(A.value * B.value)
FROM A, B
WHERE A.column_number = B.row_number
GROUP BY A.row_number, B.column_number
My confusion lies in the SUM syntax and the GROUP BY / SELECT syntax
So for my GROUP BY / SELECT confusion, I don't understand why the expressions
A.row_number and B.column_number are necessary after the SELECT statement
Why do we have to specify that when we're already using SELECT and WHERE ? To me that seems like we're saying we want to SELECT using those expressions (A.row_number and B.column_number) even though we're given back a table from WHERE already. Would it not make more sense to just say SELECT * ? I'm assuming that GROUP BY just requires you to type out the expressions it uses in the SELECT statement, but I don't know for sure.
For the SUM, I just want to clarify, the SUM is only using the A.value and the B.value from whatever is returned by the WHERE correct? Otherwise, you would be multiplying all A.value with all B.value.
Clarifying either of these would be immensely helpful. Thank you!

Answer 1

create table A
( column_number int,
row_number int,
value int
);
create table B
( column_number int,
row_number int,
value int
);
insert A (column_number,row_number,value) values (1,1,1),(1,2,2),(2,1,3),(2,2,4);
insert B (column_number,row_number,value) values (1,1,10),(1,2,20),(2,1,30),(2,2,40);
Data with your old style (non explicit) join without aggregage or group by:
SELECT A.row_number as Ar, B.column_number as Bc,
A.value as Av,B.value as Bv,A.value*B.value as product
FROM A, B
WHERE A.column_number = B.row_number
+------+------+------+------+---------+
| Ar | Bc | Av | Bv | product |
+------+------+------+------+---------+
| 1 | 1 | 1 | 10 | 10 |
| 2 | 1 | 2 | 10 | 20 |
| 1 | 1 | 3 | 20 | 60 |
| 2 | 1 | 4 | 20 | 80 |
| 1 | 2 | 1 | 30 | 30 |
| 2 | 2 | 2 | 30 | 60 |
| 1 | 2 | 3 | 40 | 120 |
| 2 | 2 | 4 | 40 | 160 |
+------+------+------+------+---------+
Seeing the above, the below gets a little more clarity:
SELECT A.row_number, B.column_number,sum(A.value * B.value) as theSum
FROM A, B
WHERE A.column_number = B.row_number
GROUP BY A.row_number, B.column_number
+------------+---------------+--------+
| row_number | column_number | theSum |
+------------+---------------+--------+
| 1 | 1 | 70 |
| 1 | 2 | 150 |
| 2 | 1 | 100 |
| 2 | 2 | 220 |
+------------+---------------+--------+

Answer 2

Giving table name after SELECT will identify which table to refer to. Mainly useful in the case where both tables have same column names.
GROUP BY will aggregate the data and display one record per grouped-by value. That is, in your case, you'll end up with only one record per row-column combination.

Answer 3

By definition multiplication of two matrices A(n,m) and B(m,p) produces a matrix C(n,p).
So the SQL for multiplication should return same data structure as was used for storage of A and B, which is three columns:
row_number
column_number
value
, with one value per (row, column) combination.
This is why you need first two in the group by clause.
WHERE clause is independent from SELECT. First is responsible for getting the right records, second for getting the right columns.