Examples
Examples
Try the starting example
- Rectangular real matrix, rows on separate lines
- 1,1 1,-1 0,0
- Full orthogonal matrix Q
- 0.707106781187, 0.707106781187, 0; 0.707106781187, -0.707106781187, 0; 0, 0, 1
- Upper-trapezoidal matrix R
- 1.41421356237, 0; 0, 1.41421356237; 0, 0
- Maximum reconstruction residual |Q R − A|
- 0
- Maximum orthogonality residual |QᵀQ − I|
- 0
Try another scenario
- Rectangular real matrix, rows on separate lines
- 1,0,1 0,2,2
- Full orthogonal matrix Q
- 1, 0; 0, 1
- Upper-trapezoidal matrix R
- 1, 0, 1; 0, 2, 2
- Maximum reconstruction residual |Q R − A|
- 0
- Maximum orthogonality residual |QᵀQ − I|
- 0
Calculation method
Calculation method
Full orthogonal matrix Q
Householder reflections build a full rows×rows orthogonal factor, including for wide or rank-deficient matrices.
Upper-trapezoidal matrix R
Apply reflections to A and choose nonnegative diagonal signs where possible. R has the original rectangular shape.
Maximum reconstruction residual |Q R − A|
Maximum entrywise factor reconstruction difference.
Maximum orthogonality residual |QᵀQ − I|
Maximum entrywise difference from the identity.
Symbols and units
- a
- Rectangular real matrix, rows on separate lines
Use the units shown. A list keeps the order in which you enter its values. The symbol ^ means “raised to the power of”.
Assumptions & limits
Assumptions & limits
- Full Householder QR for finite real matrices with each dimension at most 20. The sign convention makes R diagonal entries nonnegative where nonzero; Q columns are not a unique mathematical factorisation. Double-precision results are approximate. Numeric tables use 12 significant digits and display components at roundoff scale (32 machine epsilons relative to their row or relevant calculation scale) as zero; scalar diagnostics retain full precision.
Are these inputs private?
These calculations run in your browser. A shared link contains the values you choose to share. Only questions submitted to the separate AI finder are sent to OpenAI.
Further reading
NumPy: QR factorisation conventions ↗
Background reading; this does not imply endorsement or independent review of SumFizz.