Calculated Result
Matrix Input Panel
Enter a matrix with columns treated as vectors. The tool builds an orthonormal basis and the matching upper triangular factor.
Matrix Outputs
Input Matrix A
Orthogonal Matrix Q
Upper Triangular Matrix R
Reconstructed Matrix Q × R
QTQ Check
Plotly Graph
Interpretation Notes
Formula Used
For a matrix A = [a1, a2, ..., an], the Gram-Schmidt process creates orthonormal vectors that form the columns of Q.
Projection step
rij = qiTaj for i < j
Orthogonal remainder
uj = aj - Σ rijqi
Normalization
rjj = ||uj|| and qj = uj / rjj
The decomposition satisfies A = QR, where Q has orthonormal columns and R is upper triangular.
How to Use This Calculator
- Enter the number of rows and columns for your matrix.
- Paste the matrix values using one row per line.
- Choose the delimiter that matches your pasted data.
- Set the precision and preferred graph mode.
- Click Calculate QR Decomposition.
- Review Q, R, reconstruction accuracy, and orthogonality checks.
- Download a CSV summary or PDF report when needed.
Example Data Table
| Example | Rows | Columns | Matrix A | Expected Insight |
|---|---|---|---|---|
| Classic square case | 3 | 3 | [12 -51 4; 6 167 -68; -4 24 -41] | Produces a stable full QR factorization. |
| Identity matrix | 3 | 3 | [1 0 0; 0 1 0; 0 0 1] | Q and R both remain identity. |
| Rectangular case | 4 | 3 | [1 1 0; 1 0 1; 0 1 1; 1 1 1] | Builds orthonormal columns for tall matrices. |
Frequently Asked Questions
1. What does QR decomposition do?
It factors a matrix into an orthonormal matrix Q and an upper triangular matrix R. This helps with least-squares problems, solving systems, and understanding column relationships in scientific models.
2. Why use Gram-Schmidt here?
Gram-Schmidt shows the decomposition process directly. It reveals each projection, subtraction, and normalization step, which makes the geometry of orthogonalization easier to inspect and teach.
3. Can the calculator handle rectangular matrices?
Yes. It supports tall matrices where columns do not exceed rows. In that case, the tool generates an orthonormal basis for the column space and the corresponding triangular factor.
4. What happens with dependent columns?
If one column depends on previous columns, its orthogonal remainder becomes nearly zero. The calculator will flag the matrix as rank-deficient because normalization would be unstable or impossible.
5. Why check QTQ?
For a correct orthonormal matrix, QTQ should match the identity matrix. Comparing them confirms how closely the computed basis satisfies orthogonality within numerical precision.
6. What does the reconstruction matrix show?
The reconstruction matrix multiplies Q and R again. It should match the original matrix A closely. Small differences usually come from rounding and floating-point arithmetic.
7. What does the graph represent?
The graph can display original column norms, the diagonal of R, or magnitudes from Q. These views help compare vector scaling, normalization strength, and the structure of the decomposition.
8. When is this useful in physics?
It is useful in model fitting, signal separation, coordinate transformations, and numerical simulation. Orthogonal bases simplify projections and improve interpretation of independent components in measured data.