Question

How do you find the `LU` factorization for `tildeA = [(1,0),(-3,1)]` such that `tildeL` is unit diagonal?

I had gotten an `LU` factorization, except my `tildeL` matrix wasn't lower-triangular, so it doesn't count. Here's the expression I used:

If `tildeA = tildeLtildeU`, then when `tildeE_ncdotcdotcdottildeE_2tildeE_1tildeA -> tildeU`:

`color(blue)(tildeA = stackrel(tildeL)(overbrace(tildeE_1^(-1)tildeE_2^(-1)cdotcdotcdottildeE_n^(-1)))tildeU)`.

where `tildeE` is an elementary matrix and `tildeE^(-1)` is its inverse. `tildeU` is an upper-triangular matrix, and `tildeL` is a lower-triangular matrix.

Answer 1

OH, I figured out what was going on. It turns out that the upper triangular matrix `tildeU` is allowed to be the identity matrix `[(1,0),(0,1)]`.

So, I already have `tildeL` apparently. Since `tildeA = tildeLtildeU` and `tildeU = tildeI`, `tildeA = tildeL`. Therefore:

`tildeL = [(1,0),(-3,1)]`
`tildeU = [(1,0),(0,1)]`

`tildeA = tildeLtildeU = [(1,0),(-3,1)][(1,0),(0,1)] = color(blue)([(1,0),(-3,1)])`

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The longer way involves elementary matrices. Since each elementary row operation corresponds to the multiplication by an elementary matrix, we can do the following:

`stackrel(tildeA)(overbrace([(1,0),(-3,1)]))stackrel(3R_1 + R_2" ")(->)[(1,0),(0,1)] = tildeU`

Thus, `stackrel(tildeI)(overbrace([(1,0),(0,1)]))stackrel(3R_1 + R_2" ")(->) [(1,3),(0,1)] = tildeE_1`

So what we have is:

`tildeE_1tildeA = tildeU`

the condition for which `tildeA = tildeE_1^(-1)tildeE_2^(-1)cdotcdotcdottildeE_n^(-1)tildeU`.

As a result, we should realize that if we undo `tildeE_1` and transform it into the identity matrix `tildeI` using the inverse, `tildeE_1^(-1)`, we have the `LU` factorization.

`tildeE_1^(-1) = [(1,-3),(0,1)]`

`tildeE_1^(-1)tildeE_1tildeA = tildeE_1^(-1)tildeU`

`tildeItildeA = tildeA = tildeLtildeU = stackrel(tildeE_1^(-1))(overbrace([(1,0),(-3,1)]))stackrel(tildeU)(overbrace([(1,0),(0,1)])) = color(blue)(stackrel(tildeA)(overbrace([(1,0),(-3,1)])))`

so `tildeA = color(blue)(tildeL = [(1,0),(-3,1)])`.