This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# MatrixPower

 MatrixPowergives the matrix power of the matrix m. MatrixPowergives the n matrix power of the matrix m applied to the vector v.
• MatrixPower effectively evaluates the product of a matrix with itself n times.
• When n is negative, MatrixPower finds powers of the inverse of the matrix m. »
Compute for a symbolic (assumed to be integer) power:
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Compute for a symbolic (assumed to be integer) power:
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 Scope   (4)
Find the second inverse matrix power applied to a particular vector:
This is a more efficient way of computing :
Use exact arithmetic to compute the matrix power:
Use machine arithmetic:
Use 20-digit precision arithmetic:
Compute the matrix power for a complex matrix:
Matrix power for a sparse matrix:
The middle row is a stencil for a second-order approximation to the sixth derivative:
Use a symbolic matrix with a symbolic power:
 Applications   (6)
Construct a rotation matrix as a limit of repeated infinitesimal transformations:
Solve the constant coefficient system of difference equations :
This computes the fundamental solution :
Show that it satisfies equations and initial conditions:
Find the matrix exponential for a matrix without a full set of eigenvectors:
Compute the exponential as the power series for each term:
Get a sparse identity matrix with size, precision, and data type consistent with an input matrix:
Sparse identity matrix with exact values:
Sparse identity matrix with machine-number values:
Inverse power iteration for the smallest eigenvalue of a sparse positive definite matrix:
Check the error in :
Shifted inverse power iteration for the largest eigenvalue:
Check the error in :
An easy way to evaluate a matrix polynomial:
Evaluate a characteristic polynomial:
If m is nonsingular, MatrixPower[m, n].MatrixPower[m, -n] is the identity: