# 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.