Last updated September 24, 2026 · Report an error
The matrix exponential e^A extends the ordinary exponential function to square matrices, using the same infinite series definition (I + A + A²/2! + A³/3! + ...) but with matrix multiplication. It's the key tool for solving systems of linear differential equations, where the solution takes the form e^(At) times an initial condition.
This calculator computes e^A for any square matrix up to 6×6 using scaling-and-squaring: A is divided down until the series converges quickly, then the result is squared back up.
How it works
Enter a square matrix. The tool scales A down by a power of 2 so its entries are small, sums a truncated Taylor series for the scaled matrix, then squares the result back up the same number of times to recover e^A.
- Click Calculate to see your results.
Examples
A diagonal matrix
For a diagonal matrix diag(1, 2), the matrix exponential is simply diag(e¹, e²) ≈ diag(2.718, 7.389), since exponentiating a diagonal matrix just exponentiates each diagonal entry independently.
Who should use it
- Solving systems of linear differential equations of the form dx/dt = Ax.
- Modeling continuous-time Markov chains or control systems.
Industry applications
- Control theory and systems engineering
- Applied mathematics and physics
Advantages
- Uses scaling-and-squaring, the standard numerically robust method (rather than a raw, slowly-converging series sum).
- Works for any square matrix up to 6×6, not just diagonal or triangular ones.
Limitations
- Results are numerical approximations (to several decimal places), not exact symbolic expressions.
Common mistakes to avoid
- Assuming e^A can be found by exponentiating each entry of A individually — this is only true when A is diagonal.
- Assuming e^(A+B) = e^A · e^B in general — this identity only holds when A and B commute (AB = BA), unlike with ordinary numbers.
Best practices
- For a diagonal matrix, sanity-check the result by comparing each diagonal entry of e^A to the ordinary exponential of the corresponding diagonal entry of A.
Tips
- For a matrix with all-zero entries, e^A is always the identity matrix — a quick way to sanity-check the tool.