PLU decomposition - An Example

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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)

Опубликовано:

29 июн 2024

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Комментарии : 45

@brightsideofmaths Год назад

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

@jaysonandre295 3 года назад

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

@PunmasterSTP 2 года назад

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

@blaise3004 2 года назад

Thanks for the nice explanation!

@yassinebesseghir2754 2 года назад

that's awesome ! thanks!!!

@marianoaponte2518 3 года назад

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!

@brightsideofmaths 3 года назад

Thanks for the comment. You are absolutely right. My algorithm is a pen-and-paper strategy and in general is not numerically stable.

@marianoaponte2518 3 года назад

@@brightsideofmaths but you made it easy, keep going ❤️

@readbhagwatgeeta3810 Год назад

@@marianoaponte2518 can you please explain in detail your point ? I am not able to get it😢

@brightsideofmaths Год назад

@@readbhagwatgeeta3810 Computers can divide perfectly because rounding errors occur. So you want to avoid too many rounding errors in your algorithm.

@darcash1738 5 месяцев назад

Where would you pivot around for it to be the max? I don’t get the terminology here.

@gonruz 6 месяцев назад

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

@brightsideofmaths 6 месяцев назад

Nice! Thank you :)

@daisy6392 2 года назад

This is such a helpful video, thank you so much

@mandyu8621 3 года назад

Thank you so much! It is really helpful!

@brightsideofmaths 3 года назад

You're welcome! :)

@consumeentertainment9310 2 года назад

Nicely explained !!

@phanhuy5350 2 года назад

thank you so much

@lukas6610 Год назад

Thanks saved me alot of time

@brightsideofmaths Год назад

Thanks for the support :)

@PaulWintz 5 месяцев назад

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.

@brightsideofmaths 5 месяцев назад

Very good. Always remember that the PLU decomposition is just the Gaussian elimination in the correct order :)

@qihanlu2612 Год назад

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?

@brightsideofmaths Год назад

What do you mean by starting at the end? You can check my video about the LU-decomposition to see the differences :)

@patrickleah4129 2 года назад

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.

@brightsideofmaths 2 года назад

What do you mean exactly?

@hiuwakwan7377 3 года назад

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

@brightsideofmaths 3 года назад

Just apply the matrix P_34 from the right. Write the calculation down and you see it :)

@hiuwakwan7377 3 года назад

@@brightsideofmaths i donno why we need to change the column and row of lower triangular matrix.

@luiskao8438 3 года назад

@@hiuwakwan7377 Matrix A has to be decomposed into the form PLU. So in the equation A = (P_12)L(P_34)U, we have to make (L(P_34)) into a new L. And because the result of (L(P_34)) will not be a lower triangular matrix, we have to switch the row 3 and 4 again.

@flov74 Год назад

What are the pros and cons between PLU and LU square matrix decomposition?

@brightsideofmaths Год назад

PLU allows permutations. This makes it easier for the calculation.

@PaulWintz 5 месяцев назад

@@brightsideofmaths It's also necessary for some matrices, such as A = [0, 1; 1, 0], which cannot be written as A = LU but can be written as PA=LU.

@brightsideofmaths 5 месяцев назад

Indeed!@@PaulWintz

@abcrm101 7 месяцев назад

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?

@brightsideofmaths 7 месяцев назад

Multiplying with P :) You can find that in my Linear Algebra course tbsom.de/s/la

@ayang315 8 месяцев назад

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

@brightsideofmaths 8 месяцев назад

I have a linear algebra course where I go into the theoretical detail of row exchanges: tbsom.de/s/la

@kelvinchandra2004 Месяц назад

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! 🙏

@brightsideofmaths Месяц назад

Multiply P^{-1} to both side from the left to get the other result.

@kelvinchandra2004 Месяц назад

@@brightsideofmaths Ok, thank you very much!

@brightsideofmaths Месяц назад

More details in my book, see link in the description.