Only if AB=BA. In general, no! Matrices do not commute.

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Matrix Exponential Calculator

Compute e^A using Taylor series for 2x2 and 3x3 matrices

Enter 2x2 Matrix

Matrix Exponential Formula

e^A = I + A + A^2/2! + A^3/3! + A^4/4! + ...

e^A = exp(A)

d/dt e^(At) = A e^(At)

e^(A+B) ≠ e^A e^B unless AB=BA

The matrix exponential is defined by the Taylor series e^A = I + A + A²/2! + A³/3! + ... It's used to solve systems of linear ODEs and is fundamental in differential equations, control theory, and quantum mechanics.

⚠ Matrix exponential is only defined for square matrices. e^(A+B) ≠ e^A e^B in general!

What is Matrix Exponential?

The matrix exponential generalizes the scalar exponential function to matrices. It's defined by the same Taylor series but with matrix powers. e^A maps A to A to A A to A to to A A to A to A to A to A.

Taylor Series

e^A = sum_{k=0}^∞ A^k / k! Infinite sum.

ODE Solution

dx/dt = A x has solution x(t) = e^(At) x(0).

Not Commuting

e^(A+B) ≠ e^A e^B unless AB=BA.

Diagonal A

If A=diag(λi), then e^A=diag(e^λi).

Teaching Example: A = [[0,1],[-1,0]]
1. e^A = [[cos1, sin1],[-sin1, cos1]] = rotation matrix
2. e^(At) rotates by angle t

Applications

Differential Equations Control Theory Quantum Mechanics Physics Engineering

Frequently Asked Questions

What is matrix exponential?▼

e^A = I + A + A^2/2! + A^3/3! + ... Taylor series

e^(A+B) = e^A e^B?▼

Only if AB=BA. In general, no! Matrices don't commute.

How to compute e^A?▼

Taylor series, eigenvalues, or Padé approximation.

e^0 = I?▼

Yes, zero matrix exponentiated is identity matrix.

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