George Herriman's Krazy Kat is lounging under an umbrella.

Linear Algebra Calculators

QDR Factorization


This calculator uses Wedderburn rank reduction to find the QDR factorization of a matrix $A$. The process constructs the three matrices $Q$, $D$, $R$ in stages.

At each stage you'll have an equation $A=QDR+B$ where you start with $Q$, $D$, $R$ nonexistent, and with $B=A$. Also note that at each stage after the first $D$ is a diagonal matrix and $Q^TQ=D^{-1}$.

Now with these updated $Q$, $D$, $R$ update $B=A-QDR$. (The updated $B$ can also be obtained by subtracting $qdr$ from the previous $B$.)

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

At this point (if you've been following the Gram-Schmidt Algorithm) $R$ is upper triangular, and the columns of $Q$ contain an orthogonal basis for the column space of $A$.

If you've been following the Gram-Schmidt Algorithm with Column Pivoting, then $R$ is a (column) permuted upper triangular matrix, and the columns of $Q$ again contain an orthogonal basis (almost certainly different) for the column space of $A$.

Gram-Schmidt Algorithm
Choose $q$ to be the first nonzero column of $B$.
Gram-Schmidt Algorithm with Column Pivoting
Choose $q$ to be the column of $B$ with the largest 2-norm.


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.

Update $Q$, $D$, $R$ and $B$.

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


Back to calculator page or home page.
Krazy Kat is singing.