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# MatrixExp

 MatrixExp[m]gives the matrix exponential of m. MatrixExp[m, v]gives the matrix exponential of m applied to the vector v.
• MatrixExp[m] effectively evaluates the power series for the exponential function, with ordinary powers replaced by matrix powers.
Exponential of a 2×2 matrix:
Exponential applied to a vector:
Exponential of a 2×2 matrix:
 Out[1]=
 Out[2]=

Exponential applied to a vector:
 Out[1]=
 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:
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 :
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:
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