Examples
Examples
Try the starting example
- Rectangular real matrix, rows on separate lines
- 3,0 0,2
- Relative effective-rank threshold
- 1e-12
- Singular values in descending order
- 3, 2
- Thin left singular vectors U
- 1, 0; 0, 1
- Thin right singular vectors transposed Vᵀ
- 1, 0; 0, 1
- Effective rank at selected threshold
- 2
- Maximum reconstruction residual |U Σ Vᵀ − A|
- 0
Try another scenario
- Rectangular real matrix, rows on separate lines
- 0,2 3,0
- Relative effective-rank threshold
- 1e-12
- Singular values in descending order
- 3, 2
- Thin left singular vectors U
- 0, 1; 1, 0
- Thin right singular vectors transposed Vᵀ
- 1, 0; 0, 1
- Effective rank at selected threshold
- 2
- Maximum reconstruction residual |U Σ Vᵀ − A|
- 0
Calculation method
Calculation method
Singular values in descending order
Compute a real SVD, retaining min(rows,columns) singular values.
Thin left singular vectors U
Use the thin U; orient each paired U/V column so its largest-magnitude U component is nonnegative.
Thin right singular vectors transposed Vᵀ
Transpose the matching thin V factor.
Effective rank at selected threshold
Count singular values strictly above largest×threshold.
Maximum reconstruction residual |U Σ Vᵀ − A|
Reconstruct using full internal precision and report the largest entrywise absolute error.
Symbols and units
- a
- Rectangular real matrix, rows on separate lines
- b
- Relative effective-rank threshold
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
- Real thin SVD for tall, square or wide matrices up to 20 per dimension. Repeated singular values permit nonunique singular-vector bases; rank depends on the selected threshold. 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
ml-matrix: numerical decompositions ↗
Background reading; this does not imply endorsement or independent review of SumFizz.