Built-in Mathematica Symbol

MatrixPower

MatrixPower[m, n]
gives the n

matrix power of the matrix m.

MatrixPower[m, n, v]
gives the n

matrix power of the matrix m applied to the vector v.

MORE INFORMATION

MatrixPower[m, n] 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. »

MatrixPower works only on square matrices.

MatrixPower can be used on SparseArray objects. »

EXAMPLES

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Basic Examples (3)

Compute for a symbolic (assumed to be integer) power:

Compute for a symbolic (assumed to be integer) power:

In[1]:=

Out[1]=

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:

Generalizations & Extensions (1)

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 m.v - val v:

Shifted inverse power iteration for the largest eigenvalue:

Check the error in m.v - val v:

An easy way to evaluate a matrix polynomial:

Evaluate a characteristic polynomial:

Properties & Relations (1)

If m is nonsingular, MatrixPower[m, n].MatrixPower[m, -n] is the identity: