Matrix Calculator

1. Definition & formulas

The matrix exponential of an $n\times n$ matrix $A$ is defined by the power series (which always converges):

Matrix exponential is fundamental for solving linear systems of ODEs $\dot{x}=Ax$, where $x(t)=e^{tA}x(0)$.

Two useful computation methods:

• Diagonal (or diagonalizable) case: if $A = VDV^{-1}$ with $D$ diagonal, then \[ e^{A} = V e^{D} V^{-1}, \quad e^{D} = \operatorname{diag}(e^{d_{11}},e^{d_{22}},\dots). \]
• Simpler special matrices: for a 2×2 rotation-like matrix $\begin{pmatrix}0 & -\theta\\ \theta & 0\end{pmatrix}$, $e^{A}$ yields a rotation matrix with $\cos,\sin$ entries. For nilpotent matrices ($N^{m}=0$) the series truncates.

2. Worked examples

Example 1 — diagonal matrix (easy)

Compute $e^{A}$ for

Example 2 — 2×2 rotation-like matrix

Compute $e^{A}$ for

3. Common mistakes

• Treating $e^{A}$ like elementwise exponential: $e^{A}\neq (e^{a_{ij}})$ in general — must use series or decomposition.
• Assuming $e^{A+B}=e^{A}e^{B}$: this is false unless $A$ and $B$ commute ($AB=BA$).
• Forgetting series convergence vs numerical truncation: practical computation truncates the series or uses diagonalization, Padé approximants, or scaling-and-squaring algorithms.
• Applying diagonalization without checking: $A$ must be diagonalizable to use $V e^{D} V^{-1}$ directly; otherwise use Jordan form or other methods.
• Mishandling complex/imaginary values: eigenvalues can be complex — exponentials will use complex exponentials (equivalently cos/sin for imaginary pairs).

4. Practice problems

Try these problems. Click Show Answer to reveal the result.

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Matrix exponentials also connect with matrix rank, SVD, and RREF.