Example of matrix factorization

If a matrix m-by-n matrix A has rank r, we see by way of example how to write it as A=XY where X is m-by-r and Y is r-byn.

[tongluo]

Great concept explained clearly and in such a short time

[grayboywilliams]

This reminds me of a Fourier transform. We are essentially decomposing the “columns” into their most important contributors.

[rodney]

I'm a little confused about the goal with matrix factorization. You mention at the beginning that it can be used for compression, but the resultant two matrices, X and Y, contain more elements than A. Thanks!

[sawrangpatesounds]

Thank you so much, this makes it so much more clear!

[lewingtonn]

mate this video is amazing and so are you, thanks so much!

[gabrielcastelo]

Very elegant and simple explanation! Thank you very much!

[amitpadgaonkar]

Why is this done? Even in this example we have X consisting of 6 elements and Y of 8 elements. That means even to store the 2 matrices requires more space than the original

[SamVidovich]

Is it sometimes possible to further factor X and Y for greater compression? If so, when?

[farenrai]

can you explains more A1=1.A1+O.A2 how you calculate can you elaborate how you write that equation ?

[thelastcipher]

what if all columns of A are linearly independent?

[Skandawin]

Also, what if you are not able to express the columns as linear combinations of A1 and A3?

[arunabhSiddhartha]

Great! It would be great if you do a tutorial on NMF.

[create_space]

Thank you! it was exactly what I needed

[MuhammadImran-uycc]

sir this is very nice explanation. sir i need to know that is it same thing as minimal rank decomposition of a matrix?
and sir, if it is so then at what condition we can have minimal rank decomposition ?
thanks in advance

[crajakumar]

Very nice explanation. Is there any matlab code for this sir

[vikaskumarperumala]

can anyone explain how he was taken that A1, A2, A3, A4?