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Linear Algebra Calculators

LDU Factorization

This calculator uses Wedderburn rank reduction to find the LDU factorization of a matrix $A$. The process constructs the three matrices $L$, $D$, $U$ in stages. $L$ is constructed a column at a time, $D$ (a diagonal matrix) is constructed a diagonal entry at a time, and $U$ is constructed a row at a time.

At each stage you'll have an equation $A=LDU+B$ where you start with $L$, $D$, $U$ nonexistent, and with $B=A$.

Choose an entry $\beta$ in $B$ as described below.
The next column of $L$ is $c$, the column of $B$ that contains the entry $\beta$.
The next row of $U$ is $r$, the row of $B$ that contains the entry $\beta$.
The next diagonal entry of $D$ is $d=1/\beta$.

Now with these updated $L$, $D$, $U$ update $B=A-LDU$. (The updated $B$ can also be obtained by subtracting $cdr$ from the previous $B$.)

Eventually $B=0$ and $A=LDU$.

The only difference between this factorization and the usual LU factorization is that the $L$ in the LU factorization is the product $LD$ for the $L$ and $D$ found here.

Gaussian Elimination: Choose $\beta$ to be the first (from the top) nonzero element in the first (from the left) nonzero column of $B$.
Gaussian Elimination with Partial Pivoting: Choose $\beta$ to be the largest (in absolute value) element in the first (from the left) nonzero column of $B$.
Gaussian Elimination with Complete Pivoting: Choose $\beta$ to be 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 $L$, $D$, $U$ and $B$.

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

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