Matrix Exponential — Definition, Formula & Examples

The matrix exponential

e^A

extends the ordinary exponential function to square matrices. It is defined as an infinite series and is the key tool for solving systems of linear differential equations with constant coefficients.

For an $n \times n$ matrix $A$ , the matrix exponential is defined by the convergent power series $e^A = \sum_{k=0}^{\infty} \frac{A^k}{k!}$ , where $A^0 = I$ (the identity matrix). This series converges for every square matrix $A$ over $\mathbb{R}$ or $\mathbb{C}$ .

Key Formula

e^{A} = \sum_{k=0}^{\infty} \frac{A^k}{k!} = I + A + \frac{A^2}{2!} + \frac{A^3}{3!} + \cdots

Where:

$A$ = An n × n square matrix
$I$ = The n × n identity matrix (= A⁰)
$k$ = Summation index running from 0 to infinity

How It Works

To compute

e^A

in practice, you rarely sum the series term by term. For diagonalizable matrices, write

A = PDP^{-1}

where

D

is diagonal; then

e^A = P\, e^D\, P^{-1}

, and

e^D

is simply the diagonal matrix with entries

e^{d_i}

. For non-diagonalizable matrices, you use the Jordan normal form instead. The matrix exponential satisfies

\frac{d}{dt} e^{tA} = A\, e^{tA}

, which is why the general solution to

\mathbf{x}' = A\mathbf{x}

with initial condition

\mathbf{x}(0) = \mathbf{x}_0

\mathbf{x}(t) = e^{tA}\mathbf{x}_0

Worked Example

Problem: Compute the matrix exponential

e^A

where

A = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}

replaced by a more interesting case:

A = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}

Recognize the structure: A is already a diagonal matrix with diagonal entries 1 and 2.

A = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}

Apply the exponential to each diagonal entry: For a diagonal matrix, the matrix exponential is the diagonal matrix of the exponentials of the entries.

e^A = \begin{pmatrix} e^1 & 0 \\ 0 & e^2 \end{pmatrix}

Write the result: Evaluate the scalar exponentials.

e^A = \begin{pmatrix} e & 0 \\ 0 & e^2 \end{pmatrix} \approx \begin{pmatrix} 2.718 & 0 \\ 0 & 7.389 \end{pmatrix}

Answer:

e^A = \begin{pmatrix} e & 0 \\ 0 & e^2 \end{pmatrix}

Why It Matters

The matrix exponential is central to solving systems of linear ODEs such as

\mathbf{x}' = A\mathbf{x}

, which arise in physics, engineering, and control theory. It also appears in Lie group theory, quantum mechanics (the time-evolution operator

e^{-iHt/\hbar}

), and network analysis for computing diffusion on graphs.

Common Mistakes

Mistake: Assuming

e^{A+B} = e^A e^B

for all matrices.

Correction: This identity holds only when

A

and

B

commute (

AB = BA

). In general,

e^{A+B} \neq e^A e^B

. Use the Baker–Campbell–Hausdorff formula or other methods for non-commuting matrices.