Compute e^A using Taylor series for 2x2 and 3x3 matrices
Enter 2x2 Matrix
Enter 3x3 Matrix
Result
Step-by-Step Derivation
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
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