Subspace Modeling Foundations
Projection matrices map vectors onto a target subspace while minimizing orthogonal error. In this calculator, users define a basis matrix A and generate P = A(ATA)-1AT. This process is central to numerical linear algebra, where stable subspace representation improves interpretability. The generated matrix can be reused for many vectors, making repeated calculations efficient in coursework, simulation studies, and analytical pipelines.
Interpretation of Matrix Components
The matrix A contains basis vectors as columns, and ATA is the Gram matrix describing pairwise inner products. Its inverse exists when the basis columns are linearly independent. The calculator displays these intermediate matrices so users can inspect structure, scale, and conditioning. This visibility helps learners understand why dependent vectors cause singularity and why full column rank is required for the standard orthogonal projection formula.
Projected Vector and Error Insights
When a vector b is provided, the calculator computes Pb, the closest vector to b inside the column space of A. This result is useful in least squares interpretation, signal approximation, and geometric decomposition. Users can compare b with Pb to study residual behavior. In professional settings, this supports data fitting tasks where model components represent constrained directions or engineered basis features.
Validation Checks for Reliability
A valid orthogonal projection matrix is symmetric and idempotent. The tool verifies both conditions numerically by reporting maximum absolute differences for P2 − P and PT − P. Very small values confirm consistency, while larger deviations may indicate rounding issues or malformed inputs. These checks add confidence for technical reporting, lab exercises, and internal documentation where matrix correctness must be demonstrated clearly.
Documentation and Export Workflow
The calculator includes CSV and PDF exports so outputs can move directly into notebooks, spreadsheets, or audit files. The exported data captures dimensions, matrix values, and verification metrics in a structured format. Combined with the example table and formula section, this supports a professional workflow for teaching, engineering review, and quantitative analysis, especially when reproducible matrix calculations are required across multiple scenarios. This also supports benchmarking projection behavior across sample datasets, comparing basis choices, and documenting numerical assumptions during peer review cycles for teams.
FAQs
1) What does this calculator generate?
It generates the orthogonal projection matrix for the column space of your basis matrix A, and it can also project an optional vector b onto that same subspace.
2) Why do I get a singular matrix error?
The basis columns are likely linearly dependent, so ATA cannot be inverted. Remove redundant columns or choose an independent basis before recalculating.
3) Can I use this for least squares understanding?
Yes. The projected vector Pb is the fitted component in the column space, which matches the geometric interpretation used in least squares regression.
4) What do the property checks mean?
They test whether the computed matrix behaves like a projection matrix. A correct orthogonal projection is symmetric and idempotent, so both errors should be near zero.
5) What input format should I use?
Enter matrix rows on separate lines, using commas or spaces between values. The optional vector can be entered line by line or as one comma-separated list.
6) Are exports included in the results?
Yes. After calculation, you can export a CSV table or PDF summary containing dimensions, matrix outputs, projected vector values, and verification metrics.