Linear and multilinear algebra; matrix theory Definition

Matrix exponential

Definition

Let $\mathbb{K}$ be $\R$ or $\C$ , and let $A$ be an $n \times n$ matrix over $\mathbb{K}$ . The matrix exponential of $A$ is the $n \times n$ matrix $e^A = \exp(A) = \sum_{k=0}^{\infty} \frac{1}{k!} A^k = I_n + A + \frac{1}{2} A^2 + \frac{1}{6} A^3 + \cdots,$ with $A^0 = I_n$ .

Remarks

The series converges for each matrix $A$ , entry by entry. For a diagonal matrix $D$ with entries $d_i$ , $e^D$ is diagonal with entries $e^{d_i}$ . If $A = P D P^{-1}$ , then $e^A = P e^D P^{-1}$ . The function $x(t) = e^{tA} x_0$ solves the linear differential equation $x' = Ax$ with $x(0) = x_0$ .

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Matrix definition
Square matrix definition
Matrix exponential definition

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Tag	`0K665H5V1S`
URI	`m2://0K665H5V1S`
Kind	definition
Writing	W1 stated
MSC	15A16
Defines	Matrix exponential
Uses	Square matrix

Matrix exponential

Definition

Remarks

Reading path

Where this leads

Further reading

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