Formula used
The matrix exponential is defined by the convergent series:
For better numerical stability, the default method uses scaling and squaring with a Padé rational approximation:
- Scale: A ← A / 2s so that ‖A‖ is small.
- Approximate: exp(A) ≈ (V−U)−1(V+U) using order 13 Padé polynomials.
- Square back: exp(original A) = exp(A)2s.
How to use this calculator
- Paste a square matrix into Matrix input, one row per line.
- Set t to compute exp(tA) (use 1 for exp(A)).
- Keep the recommended method, or choose Taylor for study.
- Click Compute exp(tA). The result appears above the form.
- Use Download CSV or Download PDF to export the output.
Example data table
Example uses A = [[0, −1], [1, 0]] and t = 1 (a rotation generator). Expected exp(A) is a 2D rotation matrix.
| Input matrix A | Computed exp(A) (approx) | Interpretation |
|---|---|---|
| 0 -1 1 0 |
0.540302 -0.841471 0.841471 0.540302 |
Rotation by 1 radian in the plane. |
What exp(tA) represents in practice
Matrix exponentials convert a constant coefficient system x′=Ax into x(t)=exp(tA)x(0). For stable systems, eigenvalues with negative real parts drive decay; positive real parts create growth. With A=[[0,-1],[1,0]] and t=1, exp(tA) becomes a rotation by 1 radian, matching cos(1)≈0.5403 and sin(1)≈0.8415.
Default algorithm and accuracy targets
The calculator’s recommended path uses scaling and squaring with a (13,13) Padé approximant. It scales tA by 2^s until the 1‑norm is below θ13≈5.3719, builds a rational approximation (V−U)⁻¹(V+U), then squares the result s times. For n≤8, the work is a handful of matrix multiplies plus one inversion, about O(n³). Typical s values fall between 0 and 6, and the tool can display s.
Input limits and performance guidance
For interactive reliability, matrices are limited to 1×1 through 8×8. Precision can be set from 2 to 12 decimals for display and exports. The Taylor option uses 3 to 80 terms; it is educational but may need higher terms when ‖tA‖ is large. Large entries or very large |t| can overflow any floating‑point method, so rescaling A or t is practical.
Interpreting diagnostics and sanity checks
Diagnostics show ‖tA‖₁, trace(tA), and det(tA) to help spot mistakes. The 1‑norm shown is the maximum column sum, useful for comparing scaling thresholds across inputs quickly visually. A sudden jump in ‖tA‖₁ often indicates a pasted row shift or missing delimiter. For diagonal A, exp(tA) should be diagonal with exp(t a_ii) entries. A strong check is det(exp(tA))≈exp(trace(tA)); gaps usually mean rounding or ill‑conditioning. If A is nilpotent (A^k=0), the series terminates exactly after k−1 powers.
Exports and reporting workflow
After computation, export the result as CSV for spreadsheets or as PDF for reports. CSV preserves numeric structure for downstream simulation, while PDF is suited for documentation and peer review. Use the copy feature to paste tab‑delimited matrices into notebooks, then cite the method (Padé 13 or Taylor) and the chosen precision.
FAQs
1. What does this calculator compute?
It computes exp(tA), the matrix exponential of a square matrix A scaled by time t. This maps x(0) to x(t) for linear systems and supports rotations, growth, decay, and coupled dynamics.
2. Why is the Padé method recommended?
Scaling and squaring with a Padé (13,13) approximation is a standard, stable approach. It reduces the matrix magnitude before approximation, then squares back, typically giving accurate results with fewer terms than a plain series.
3. What matrix sizes are supported?
You can enter matrices from 1×1 up to 8×8. This keeps computations fast in a browser-style workflow while still covering many control, physics, and modeling tasks.
4. When should I try the Taylor series option?
Use Taylor series for learning, for very small ‖tA‖, or when you want to observe convergence by increasing terms. If results change noticeably as terms rise, the series needs more terms or the Padé method is safer.
5. How can I sanity-check the output?
Check special structures: diagonal A stays diagonal with exp(t a_ii) on the diagonal. Also compare det(exp(tA)) with exp(trace(tA)); large gaps can indicate rounding, extreme scaling, or an ill-conditioned inverse step.
6. What do the CSV and PDF downloads include?
CSV contains the last computed exp(tA) matrix as numeric rows and columns for spreadsheets or simulation. PDF includes the method label, key diagnostics, and a monospaced matrix block for reporting and sharing.