Linear and multilinear algebra; matrix theory Theorem

Exponential of a sum of commuting matrices

Statement

Let $\mathbb{K}$ be $\R$ or $\C$ , and let $A$ and $B$ be $n \times n$ matrices over $\mathbb{K}$ with $AB = BA$ . Then $e^{A + B} = e^A e^B,$ where $e^A$ is the matrix exponential.

Remarks

The proof multiplies the two series and uses the binomial formula, which needs $AB = BA$ . It gives $e^A e^{-A} = I_n$ , so $e^A$ is always invertible. Without $AB = BA$ , the identity can fail, for example for $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$ .

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Matrix definition
Square matrix definition
Matrix exponential definition
Exponential of a sum of commuting matrices statement

Tag	`04BD23PSK5`
URI	`m2://04BD23PSK5`
Kind	statement, theorem
Writing	W0 stub
MSC	15A16
Status	This copy of the site has no build data. Run `m2 status 04BD23PSK5`.
Uses	Matrix exponential

Exponential of a sum of commuting matrices

Statement

Remarks

Reading path

Further reading

What this page needs for W1

What links here