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Thread: Cholesky Decomposition

  1. #1
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    Cholesky Decomposition

    Hi, I'm currently taking linear algebra and my professor gave us a homework on Cholesky decompositions. However, when I look at his solution, I don't understand what he's doing when he is doing iterate. I understand the first iteration where he is calculating the L1 column, but I don't understand how he is getting S in the next iterations. Here is his solution:
    Cholesky Decomposition-cholesky.png

    Thank you so much for the help!
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  2. #2
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    Re: Cholesky Decomposition

    Looking at this I see that

    $L_1 = \dfrac {1}{\sqrt{S_{1,1}}} \text{ 1st column of S} = \begin{pmatrix}2 \\ -3 \\ 4 \end{pmatrix}$

    Then they are using $L_1$ to zero out entries in columns 2 and 3 of S and then looking that the 2x2 matrix with row and column 1 removed.

    in other words

    the second column of $S$ has $3 L_1$ added to it to become $\begin{pmatrix}0 \\ 4 \\ -4 \end{pmatrix}$

    the third column of $S$ has $-4 L_1$ added to it to become $\begin{pmatrix}0 \\ -4 \\ 20 \end{pmatrix}$

    and now we start over using the lower 2 rows and columns of the resulting matrix above, i.e. $S=\begin{pmatrix}4 &-4 \\ -4 &20 \end{pmatrix}$

    Now repeat all the above steps

    $L_2 = \dfrac {1}{\sqrt{S_{1,1}}} \text{ 1st column of S} = \begin{pmatrix} 2 \\ -2 \end{pmatrix}$

    the second column of $S$ has $2 L_2$ added to it to become $\begin{pmatrix}0 \\ 16 \end{pmatrix}$

    and again we take $S$ to be the reduced matrix formed by removing the first row and column. Thus

    $S=16$

    and $L_3 = \dfrac{1}{\sqrt{S_{1,1,}}} \text{ 1st column of S} = 4$
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  3. #3
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    Re: Cholesky Decomposition

    Thank you so much! If I may ask, how did you figure that out? Did you already know this method of solving it because I can't find this method online.
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  4. #4
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    Re: Cholesky Decomposition

    Quote Originally Posted by STEMSTRUGGLES View Post
    Thank you so much! If I may ask, how did you figure that out? Did you already know this method of solving it because I can't find this method online.
    I just looked at what they had done. I didn't realize about the square root until the last term when they got 4 from 16.
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