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)

Пікірлер: 45

@brightsideofmaths Жыл бұрын

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

@PunmasterSTP 2 жыл бұрын

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

@jaysonandre295 3 жыл бұрын

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

@blaise3004 2 жыл бұрын

Thanks for the nice explanation!

@yassinebesseghir2754 2 жыл бұрын

that's awesome ! thanks!!!

@marianoaponte2518 3 жыл бұрын

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 жыл бұрын

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 жыл бұрын

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

@readbhagwatgeeta3810 Жыл бұрын

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

@brightsideofmaths Жыл бұрын

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

@darcash1738 6 ай бұрын

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

@phanhuy5350 2 жыл бұрын

thank you so much

@mandyu8621 3 жыл бұрын

Thank you so much! It is really helpful!

@brightsideofmaths 3 жыл бұрын

You're welcome! :)

@hiuwakwan7377 3 жыл бұрын

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

@brightsideofmaths 3 жыл бұрын

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

@hiuwakwan7377 3 жыл бұрын

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

@luiskao8438 3 жыл бұрын

@@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.

@daisy6392 2 жыл бұрын

This is such a helpful video, thank you so much

@gonruz 7 ай бұрын

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 7 ай бұрын

Nice! Thank you :)

@qihanlu2612 2 жыл бұрын

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 Жыл бұрын

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

@consumeentertainment9310 2 жыл бұрын

Nicely explained !!

@lukas6610 Жыл бұрын

Thanks saved me alot of time

@brightsideofmaths Жыл бұрын

Thanks for the support :)

@patrickleah4129 2 жыл бұрын

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 жыл бұрын

What do you mean exactly?

@PaulWintz 6 ай бұрын

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 6 ай бұрын

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

@abcrm101 8 ай бұрын

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 8 ай бұрын

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

@flov74 Жыл бұрын

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

@brightsideofmaths Жыл бұрын

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

@PaulWintz 6 ай бұрын

@@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 6 ай бұрын

Indeed!@@PaulWintz

@ayang315 9 ай бұрын

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 9 ай бұрын

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

@kelvinchandra2004 3 ай бұрын

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 3 ай бұрын

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

@kelvinchandra2004 3 ай бұрын

@@brightsideofmaths Ok, thank you very much!

@brightsideofmaths 3 ай бұрын

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