Linear Algebra 13b: The Effect of an Elementary Matrix on the Other Matrix in a Product

[raykinane]

Wonderful fun presentation. Thank you :)

[worldfacade]

Another way of coming at elementary matrices that I find helpful is to start by posing the question:

I wonder, does there exist a matrix E such that, when it multiplies any other matrix, has the same effect as performing a row/column operation? If there does, what would it look like?

Well, suppose E exists. E = EI, because when I acts on E it leaves it the same, but looking at it from another perspective we might ask what does E do to I? Ex hypothesi it must perform the desired row/column operation on it. So, if such a matrix as E exists, it must look like the identity with the desired operation performed on it.

That doesn't explain why these matrices do exist if you want to multiply on the correct side for the type of operation (row or column) you want, but not on the other side. Still, it helps to see (what wasn't obvious to me at first) that IF they exist then they MUST look like what you have been describing.

[ahmedhemani]

On you tube I see videos labelled 15nn onwards. What about 14nnn videos ? I am looking for modified Gramm Schmidt, householder xformation and givens rotation as the methods for orthognalizing matrices. Which are the right set of videos. Thanks for in depth presentation.

[arcana]

Umm.. I think it can be proven easily by induction...