This generator builds diagonal scaling matrices for geometry, graphics, linear algebra, and coordinate transforms. It also reports determinant behavior, inverse availability, eigenvalues, and optional vector scaling results.
Scaling Matrix Generator Form
Example Data Table
| Case | Dimension | Scale factors | Input vector | Determinant | Scaled vector |
|---|---|---|---|---|---|
| Planar resize | 2D | 2, 0.5 | (3, 8) | 1 | (6, 4) |
| Signed 3D scaling | 3D | 2, 0.5, -1.5 | (4, -2, 1) | -1.5 | (8, -1, -1.5) |
| Degenerate map | 4D | 1, 3, 0, 2 | (1, 1, 1, 1) | 0 | (1, 3, 0, 2) |
Formula Used
For a scaling transform, the generated matrix is diagonal. Each diagonal entry represents the scale factor along one coordinate axis.
The determinant gives the signed area, volume, or hypervolume multiplier. Its absolute value shows overall size change. Its sign shows whether orientation is preserved or flipped.
The inverse exists only when every scale factor is nonzero. A zero factor collapses at least one dimension, so the transformation becomes singular.
To scale a vector, multiply the matrix by the coordinate vector. Because the matrix is diagonal, each component simply multiplies by its matching scale factor.
When homogeneous mode is enabled, the tool adds a final one on the diagonal. This format is useful when scaling is combined with affine transformations.
How to Use This Calculator
- Choose the number of dimensions you need, from 2D through 6D.
- Enter one scale factor for each active axis. Positive, negative, fractional, and zero values are allowed.
- Set your preferred decimal precision and choose fixed or scientific notation.
- Optionally enter a vector with matching coordinates to see the transformed output immediately.
- Enable homogeneous mode when you want an added trailing one for affine coordinate workflows.
- Press Generate Matrix to display the result above the form under the page header.
- Review determinant, rank, trace, eigenvalues, inverse availability, and axis-by-axis effects.
- Use the export buttons to save the visible result as CSV or PDF.
Frequently Asked Questions
1. What does this calculator generate?
It creates a diagonal scaling matrix from your selected axis factors. It also reports determinant, trace, rank, eigenvalues, singular values, inverse availability, and optional transformed vector coordinates.
2. What does a negative scale factor mean?
A negative factor reflects the object across that axis while also scaling its size. If an odd number of factors are negative, the transformation flips orientation overall.
3. What happens when a scale factor is zero?
That axis collapses to zero length. The determinant becomes zero, the rank drops, and the inverse no longer exists because information is lost during the transformation.
4. Why are the eigenvalues equal to the scale factors?
For a diagonal scaling matrix, the diagonal entries are the eigenvalues. Each axis basis vector stays aligned with itself, so its scaling amount becomes the corresponding eigenvalue.
5. What is the determinant telling me?
The determinant measures the signed size multiplier. In 2D it scales area, in 3D it scales volume, and in higher dimensions it scales hypervolume.
6. When should I use homogeneous mode?
Use it when your workflow combines scaling with affine operations such as translation in graphics pipelines. The extra trailing one keeps coordinates compatible with homogeneous transformations.
7. How should I enter the vector values?
Enter comma-separated numbers matching the chosen dimension exactly. For example, use “3, -1” for 2D or “4, 0, 2” for 3D coordinates.
8. Can I export the matrix and summary?
Yes. After generating a result, use the CSV or PDF buttons above the form. They export the visible summary, axis effects, and matrix tables.