# How can I use this Cholesky decomposition algorithm on this example?

In this course, the authors introduce a method for Cholesky decomposition of matrix $$A$$, based on row reduction:

Procedure 7.4.1: Finding the Cholesky Factorization

1. Using only type 3 elementary row operations (multiples of rows added to other rows) put A in upper triangular form. Call this matrix $$\hat U$$. Then $$\hat U$$ has positive entries on the main diagonal.
2. Divide each row of $$\hat U$$ by the square root of the diagonal entry in that row. The result is the matrix $$U$$.

Example:

$$\begin{bmatrix} 9 & -6 & 3 \\ -6 & 5 & -3 \\ 3 & -3 & 6 \\ \end{bmatrix} \rightarrow \begin{bmatrix} 9 & -6 & 3 \\ 0 & 1 & -1 \\ 0 & -1 & 5 \\ \end{bmatrix} \rightarrow \begin{bmatrix} 9 & -6 & 3 \\ 0 & 1 & -1 \\ 0 & 0 & 4 \\ \end{bmatrix}$$

• I wasn't able to find this algorithm on other sites, which seems strange,
• I don't understand how this algorithm is used on the example provided.

For example, which row operations (of type 3) can change the second row from $$[-6, 5, -3]$$ to $$[0, 1, -1]$$? Adding 2xR3 to R2 gives $$[0, -1, 9]$$ and from there I cannot figure out how this can be improved.

Help appreciated, thanks.

• When creating upper triangle, you only use the row(s) above the one you’re working with. In this case add $2/3$ of the first row to the second Jan 5 at 13:34

There is a mistake in the algorithm: the allowed operations must be adding a multiple of a row to other rows below it. Only these elementary operations are implemented by lower triangular matrices with $$1$$ on the diagonal, and in this case we will indeed obtain the $$U$$ part of $$LU$$ decomposition, from which it is possible to get Cholesky decomposition.
In the example we add $$\frac23$$ of the first row to the second and subtract $$\frac13$$ of the first row from the third.