# Linear Algebra Calculators

## LU Factorization

This calculator uses Wedderburn rank reduction to find the LU factorization of a matrix `A`. The process constructs the two matrices `L` and `U` in stages.

At each stage you'll have an equation `A=LU+B` where you start with `L` and `U` nonexistent and with `B=A`. The next column of `L` and row of `U` are chosen from `B`. (The `L` column is scaled.) Then `L`, `U` and `B=A-LU` are updated. Eventually `B=0` and `A=LU`.

At this point (if you've been doing *Gaussian Elimination*)
`U` is a row echelon form for `A`, and `L` (lower triangular)
contains a record of the row operations that you would have
used in row reducing `A` to `U` the standard way.

If you've been doing *Gaussian Elimination with Partial Pivoting*, then
`L` is a (row) permuted lower triangular matrix, and if
you've been doing *Gaussian Elimination with Complete Pivoting*, then
`L` is a (row) permuted lower triangular matrix while `U`
is a (column) permuted upper triangular matrix.

- Gaussian Elimination
- Pick the first nonzero element in the first nonzero column of `B`.
- Gaussian Elimination with Partial Pivoting
- Pick the largest (in absolute value) element in the first nonzero column of `B`.
- Gaussian Elimination with Complete Pivoting
- Pick the largest (in absolute value) element in the entire matrix `B`.

Either choose a size and press this button to get a randomly generated matrix, or enter your matrix in the box below. (Look at the example to see the format.)

Matrix `A`:

Select a column number.

Select a row number.

Update `L`, `U` and `B`.

The reset button leaves the `A` matrix alone, but reinitializes `L`, `U` and `B`.

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