Is there a matrix decomposition algorithm?

Question

Algorithms

Please tell me the algorithm for decomposing a matrix into a product of transvections (elementary matrices) and one diagonal matrix.

Answer 1

Did you even try to google? :)
https://dxdy.ru/post391510.html

Answer 2

Look at the Gauss method . There, a diagonal matrix is ​​obtained by adding one row to the others and rearranging the rows. So - adding one row to another with a coefficient - this is the multiplication of the matrix by the transvection.
The only problem - you have to rearrange the lines in some places. This is where the classic puzzle about swapping two numbers without using auxiliary variables will help you:

a = a + b;
b = a - b;
a = a - b;

These three operations will swap a and b. Moreover, they already look like transvections - only the second operation is not quite the same a += b*k.
But you can change a little:

a = a + b;
b = b - a;
a = a + b;

There are three transvections and the lines are swapped, only one of them is multiplied by -1. But this multiplication does not change the Gauss method at all.
Here is your algorithm.