Householder QR

Orthogonal transformations via reflections

Algorithm

We apply a sequence of orthogonal reflection matrices $H_k$ to transform $A$ into an upper triangular matrix $R$ . Each reflection acts on all columns simultaneously, aligning the $k$ -th column with the coordinate axis $e_k$ while leaving previous columns invariant.

Step-by-Step Transform

Step 0 of 4

StartFinish

Initial matrix

A

. The columns are shown as vectors

a_1, a_2, a_3

Matrix Evolution

A \rightarrow \begin{bmatrix} 1.00 & 1.00 & 0.00 \\[0.5em] 1.00 & 2.00 & 1.00 \\[0.5em] 1.00 & 0.00 & 2.00 \end{bmatrix}

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