Derivation of Cholesky Decomposition

[BobTheLawyer]

Homework Statement

Derive Cholesky Decomposition for a 3x3 matrix

Homework Equations

IN: S is Real matrix with dimensions 3x3 and is Symmetric and semi-definite
Out: L is a Real matrix with dimensions 3x3 such that

1. S=L*L^t
2. L is lower-triangular

The Attempt at a Solution

We learned this in class, and here is what I have in my notes. Near the end, it starts not making sense, so I think I recorded something wrong, and I also don't fully get what's going on.

 

[andrewkirk]

You have miscopied the subscripts in the second last line. Replace $L_{32}$ by $L_{33}$, $L_{21}$ by $L_{22}$ and $L_{31}$ by $L_{32}$ and then it will follow from the line above.

 

[BobTheLawyer]

You have miscopied the subscripts in the second last line. Replace $L_{32}$ by $L_{33}$, $L_{21}$ by $L_{22}$ and $L_{31}$ by $L_{32}$ and then it will follow from the line above.

Thank you so much! You're a life saver. This got me so confused. QR decomposition just clicked for me about 5 minutes ago, and now I get this too!