Linear Algebra Calculators

QDR Factorization

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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. $Q$ is constructed a column at a time, $D$ (a diagonal matrix) is constructed a diagonal entry at a time, and $R$ is constructed a row at a time.

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 $Q^TQ=D^{-1}$.

• The next column of $Q$ is $q$, chosen from $B$ as described below.
• The next row of $R$ is $r=q^T B$.
• The next diagonal entry of $D$ is $d=1/q^Tq$.

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.

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Either choose a size 2 3 4 5 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 $B$.

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