PLU decomposition - An Example

Thanks to all supporters! They are mentioned in the credits of the video :)

This video is about the PLU decomposition for square matrices. I show the algorithm with the help of an example.

0:00 Introduction
1:13 Example
2:00 Row exchange
2:30 Gaussian elimination
4:20 Next row exchange
5:45 Last step

I hope that this helps students, pupils and others.

#LinearAlgebra

(This explanation fits to lectures for students in their first year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[brightsideofmaths]

Don't forget to download the pdf version for free!
tbsom.de/s/ov

[PunmasterSTP]

PLU? More like "Please and thank you!" These videos are solid gold.

[jaysonandre]

You’ve got kind heart... I suscribed.. from being a stubborn guy to a subscriber of bright side .. Thanks a lot

[marianoaponte]

If you use this algorithm in finite-precision environments (such as a computer) you should use the MAX pivot on the column, instead of an arbitrary non-zero one, in order to have a multiplier less than or equal 1, which is better because it will leave the algorithm stable and won't affect precision of results! <3

Nice video man, keep uploading!

[PaulWintz]

At 5:10, when multiplying the L matrix on both sides by a permuation matrix, it is important that the columns being swapped only have 1's on the diagonal. Imagine that you are switching two non-adjacent rows/columns, say 4 and 6. The column swap could moves the (5, 4) entry, below the diagonal, into the (5, 6) entry, above the diagonal. But, since we do this operation to columns that only have zeros except on the diagonal, the lower triangular structure is presereved.

[gonruz]

I was so confused with how the row exchanges affect the L matrix, with just 10 seconds of the video i was finally capable to understand, thank you

[mandyu]

Thank you so much! It is really helpful!

[hiuwakwan]

i am so confused about 5:06 (exchange the column of c3 and c4). would you explain deeply?

[kelvinchandra]

Hi, I have a question about the final result.

Based on my textbook, the format is written as "PA=LU"
Your final result becomes "A=PLU"
So does that mean it mean that the P is also the same either in PA = LU or A = PLU, or in PA = LU, the P shall be the inverse from A = PLU?
I also figured out that if P x P = I-Matrix (1, 0, 0; 0, 1, 0; 0, 0, 1) [This is calculated that both Ps are same and not from your examples but from random thoughts.] Which means P in inverse is exact the same as P also.

I am looking forward for your help. Thank you! 🙏

[patrickleah]

what if in the final step you did in order to put the permutation matrix in the right spot- the "L" matrix did not end up as a lower triangular matrix.

[daisy]

This is such a helpful video, thank you so much

[qihanlu]

Hi, I have a question. When we are doing the normal LU decomposition, we do the reverse order, which means we start at the end. However, when we are doing PLU decomposition, why do we start at the front?

[abcrm]

is A=PLU different from PA=LU? At my Uni we are supposed to use PA=LU, how do I convert one form to the other?

[ayang]

I didn't really get why you were able to do those final row and column exchanges to the identity matrix without doing them to the matrix U

[the_eternal_student]

I did not understand aabout the squaring and column exchange.