# MatrixExp

MatrixExp[m]

gives the matrix exponential of m.

MatrixExp[m,v]

gives the matrix exponential of m applied to the vector v.

# Details and Options • MatrixExp[m] effectively evaluates the power series for the exponential function, with ordinary powers replaced by matrix powers.
• MatrixExp works only on square matrices.
• In MatrixExp[m,v] the matrix m can be a SparseArray object.

# Examples

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## Basic Examples(2)

Exponential of a 2×2 matrix:

Exponential applied to a vector:

## Scope(4)

Use exact arithmetic to compute the matrix exponential:

Use machine arithmetic:

Use 24-digit precision arithmetic:

Find the matrix exponential of a complex matrix:

The exponential of a symbolic matrix:

Exponential of a sparse 100×100 matrix:

## Applications(2)

A system of first-order linear differential equations:

Write the system in the form with :

The matrix exponential gives the basis for the general solution:

The matrix exponential applied to a vector gives a particular solution:

The matrix s approximates the second derivative periodic on on the grid x:

A vector representing a soliton on the grid x:

Propagate the solution of using a splitting :

Plot the solution and 10 times the error from the solution of the cubic Schrödinger equation:

## Properties & Relations(4)

The matrix exponential of a diagonal matrix is diagonal:

The matrix exponential of a nilpotent matrix is a polynomial matrix:

MatrixExp[m] is always invertible, and the inverse is given by MatrixExp[-m]:

If m is diagonalizable with then :

## Possible Issues(1)

For a large sparse matrix, computing the matrix exponential may take a long time:

Computing the application of it to a vector uses less memory and is much faster:

The results are essentially the same:

## Neat Examples(1)

Introduced in 1991
(2.0)
|
Updated in 2007
(6.0)