I'm have a little trouble understanding PA=LU, I have no problems with A=LU but can't seem to figure out the Permutation matrix.(adsbygoogle = window.adsbygoogle || []).push({});

So I have summarised the process I am using let me know where it can be improved.

Step 1: Using Gaussian Elimination with partial pivoting reduce A to form a matrix U.

Step 2: The matrix L is formed by the "negative of the row reduction multiples" eg: R2=R2-(3/4)R3 then -(3/4) is an element in the the matrix L.

Now here is the first problem, as with the A=LU decomposition do these multiples remain fixed in place or do they also permute around depending on row interchanges ?

eg: say R2=R2-(3/4)R3 is the first operation than (-3/4) should go in the position of Row 2, Column 1. But, later if I need to interchange R2 with say R4 (for partial pivoting) will this effect the position of (-3/4) in the matrix L ?

Step 3: Now that you have matrix L and U, form the matrix P with the row interchanges during the process of pivoting.

For example: If during pivoting R1<->R4 and R4<->R3 than

[tex] \begin{bmatrix}

1 & 0 & 0 & 0\\

0 & 1 & 0 & 0\\

0 & 0 & 1 & 0\\

0 & 0 & 0 & 1

\end{bmatrix} [/tex]

becomes(R1<->R4)

[tex] \begin{bmatrix}

0& 0 & 0 & 1\\

0 & 1 & 0 & 0\\

0 & 0 & 1 & 0\\

1 & 0 & 0 & 0

\end{bmatrix} [/tex]

and finally (R4<->R3)

[tex]\begin{bmatrix}

0& 0 & 0 & 1\\

0 & 1 & 0 & 0\\

1 & 0 & 0 & 0\\

0 & 0 & 1 & 0

\end{bmatrix} [/tex]

The lecture notes I have are extremely complicated and involve L inverse theory which makes my head hurt and I can't find any useful resources online.

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# PA=LU decomposition (w/ Partial Pivoting)

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