exponential of matrix calculator

What is the exponential of a matrix?

The matrix exponential, written as exp(A) or e^A, extends the familiar scalar exponential to square matrices. It is defined by the infinite power series:

exp(A) = I + A + A²/2! + A³/3! + ...

Here, I is the identity matrix. This object appears naturally when solving systems of linear differential equations such as x'(t) = Ax(t), where the solution is x(t) = exp(tA)x(0).

How to use this calculator

1) Enter a square matrix

Provide your matrix in the input box using spaces or commas between numbers, and new lines between rows. The matrix must be square (same number of rows and columns).

2) Choose a scalar t

Set t if you want exp(tA). For plain exp(A), leave t = 1.

3) Click calculate

The tool returns the resulting matrix and some numerical details about the computation.

Worked mini example

For the matrix A = [[0, 1], [-2, -3]], exp(A) is used in control systems and dynamical modeling to evolve states through time. This calculator handles the heavy algebra and returns a clean numerical matrix you can copy directly into reports or code.

How the calculator computes exp(A)

Numerical method used

The script applies a practical method: scaling and squaring plus a Taylor series. In short, it scales A down to improve convergence, sums series terms, then squares the result back up.

• Compute a matrix norm to estimate scale.
• Scale matrix by a power of 2: B = A / 2^s.
• Approximate exp(B) with a finite series.
• Square s times to recover exp(A).

This is stable and accurate for many common engineering, math, and physics inputs.

Applications of matrix exponentials

• Linear ODE systems and state-space models
• Control engineering and robotics
• Markov chains and transition models
• Quantum mechanics and operator evolution
• Signal processing and network dynamics

Practical tips

• If values are very large, increase maximum series terms.
• Check units before setting t, especially in physical models.
• If you expect symmetry or structure, compare output numerically to verify model assumptions.

FAQ

Does this support non-square matrices?

No. The matrix exponential is defined for square matrices only.

Can I use decimals and negative values?

Yes. Decimals, negatives, and scientific notation (like 1e-3) are supported.

Is this exact or approximate?

Numerical approximation. For most moderate-size problems, the result is highly accurate. For extreme matrices, specialized scientific software may be preferred for maximum robustness.