 Householder QR example in two dimensions

Apply the preceding Householder reflection to the columns ofThe first column becomes $(-5,0{)}^T$ and the second becomes $(-11/5,2/5{)}^T$:Thus an orthogonal matrix multiplying an upper triangular matrix reconstructs $A$. The zero below the diagonal came from a geometric operation that preserves Euclidean norms and inner products.

Our positive-diagonal convention for QR decomposition is obtained by a sign adjustment. With $D=\mathrm{diag}(-1,1)$, putThen $Q^{T}Q=I$ and $\boxed{A=Q{R}_{+}}$. The intermediate sign chooses an orientation for a basis vector; the adjustment makes the diagonal positive.

For a larger matrix, the next Householder reflection acts only on the lower rows of the remaining columns, retaining the zeros already created. Store its defining vector and apply $z\mapsto z-2u(u^{T}z)$ instead of forming a dense reflection matrix. This links a simple mirror operation to efficient QR decomposition, including least-squares solutions.