Build scaling matrices for points, vectors, and datasets. Review determinants, inverses, traces, and transformed coordinates. Plot trends, export files, and learn each calculation confidently.
This example uses scale factors 2, 0.5, and 3 on a 3D input matrix.
| Axis | Scale Factor | P1 | P2 | P3 | Scaled P1 | Scaled P2 | Scaled P3 |
|---|---|---|---|---|---|---|---|
| X | 2 | 1 | 4 | -2 | 2 | 8 | -4 |
| Y | 0.5 | 2 | -1 | 0 | 1 | -0.5 | 0 |
| Z | 3 | 3 | 5 | 1 | 9 | 15 | 3 |
Use the example button above to load these values into the calculator instantly.
S = diag(s₁, s₂, ..., sₙ)
Y = S × X, where X is the input coordinate matrix and each column is a point or vector.
det(S) = s₁ × s₂ × ... × sₙ
S⁻¹ = diag(1/s₁, 1/s₂, ..., 1/sₙ), only when every scale factor is nonzero.
trace(S) = s₁ + s₂ + ... + sₙ and the rank equals the number of nonzero scale factors.
A scaling matrix stretches, shrinks, reflects, or collapses coordinates along selected axes. Because it is diagonal, each axis is affected independently by its matching scale factor.
A negative factor reflects the object across that axis and also scales its size by the factor’s absolute value. A factor of -1 reflects without resizing.
That axis collapses to zero, making the transformation degenerate. The determinant becomes zero, the rank drops, and the matrix no longer has an inverse.
The determinant measures total area, volume, or hypervolume scaling. Its sign also tells whether orientation is preserved or reversed by the transformation.
Scaling is uniform when every diagonal factor is the same. In that case, all axes expand or shrink equally and shape proportions remain unchanged.
Yes. Each column of the input matrix can be treated as a vector or point. The calculator scales every column using the same diagonal transformation.
A diagonal scaling matrix already places its principal action on the coordinate axes. Therefore, its eigenvalues are exactly the diagonal entries, which are the scale factors themselves.
The graph compares absolute row totals before and after scaling. It helps you see how strongly each axis changes under the selected transformation.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.