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MatrixPower
gives the n matrix power of the matrix m applied to the vector v.
Details and Options
- 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. »
- When n is not an integer, MatrixPower effectively evaluates the power series for the function, with ordinary powers replaced by matrix powers. »
- MatrixPower works only on square matrices.
Examples
open allclose allBasic Examples (4)Summary of the most common use cases
https://wolfram.com/xid/0bdo09sse-xwd5iw
https://wolfram.com/xid/0bdo09sse-g1vpp7
https://wolfram.com/xid/0bdo09sse-gpxvco
Square the inverse of a symbolic matrix:
https://wolfram.com/xid/0bdo09sse-s2vrv6
https://wolfram.com/xid/0bdo09sse-p2yv4u
https://wolfram.com/xid/0bdo09sse-i5k570
Raise a matrix to the 10th power:
https://wolfram.com/xid/0bdo09sse-fgeyhr
Notice that this is different from raising each entry to the 10th power:
https://wolfram.com/xid/0bdo09sse-z7yg6i
Compute a symbolic matrix power:
https://wolfram.com/xid/0bdo09sse-fjqltk
Scope (15)Survey of the scope of standard use cases
Basic Uses (9)
Raise a machine-precision matrix to a positive integer power:
https://wolfram.com/xid/0bdo09sse-f6rr16
Raise it to a fractional power:
https://wolfram.com/xid/0bdo09sse-dwwq18
https://wolfram.com/xid/0bdo09sse-etp19y
Raise an exact matrix to an integer power:
https://wolfram.com/xid/0bdo09sse-hml8lc
Raise it to a fractional power:
https://wolfram.com/xid/0bdo09sse-84v7wg
Raise an arbitrary-precision matrix to a negative integer power:
https://wolfram.com/xid/0bdo09sse-s9aujl
https://wolfram.com/xid/0bdo09sse-vhxry5
Raise it to an irrational power:
https://wolfram.com/xid/0bdo09sse-isvqws
Raise a symbolic matrix to an integer power:
https://wolfram.com/xid/0bdo09sse-u07efd
Raise a matrix to a symbolic power:
https://wolfram.com/xid/0bdo09sse-gk4zm8
Raising large machine-precision matrices to a power is efficient:
https://wolfram.com/xid/0bdo09sse-dkq7nk
https://wolfram.com/xid/0bdo09sse-lx8juz
Directly applying the power to a single vector is even more efficient:
https://wolfram.com/xid/0bdo09sse-rz897y
Raise a matrix with finite field elements to an integer power:
https://wolfram.com/xid/0bdo09sse-m5yk5
https://wolfram.com/xid/0bdo09sse-6dui5
Raise a CenteredInterval matrix to an integer power:
https://wolfram.com/xid/0bdo09sse-kzui0z
https://wolfram.com/xid/0bdo09sse-3yenu
Find a random representative mrep of m:
https://wolfram.com/xid/0bdo09sse-kte9zj
Verify that mpow contains MatrixPower[mrep,17]:
https://wolfram.com/xid/0bdo09sse-fxghe1
Special Matrices (6)
The result of raising a sparse matrix to a positive integer power is returned as a sparse matrix:
https://wolfram.com/xid/0bdo09sse-ocj3kf
https://wolfram.com/xid/0bdo09sse-fm3xwv
https://wolfram.com/xid/0bdo09sse-eordwx
Raising a sparse matrix to a other powers will typically produce a normal matrix:
https://wolfram.com/xid/0bdo09sse-w5fh2s
Directly apply the power of of a sparse matrix to a sparse vector:
https://wolfram.com/xid/0bdo09sse-27tx1h
https://wolfram.com/xid/0bdo09sse-u09q39
https://wolfram.com/xid/0bdo09sse-luf4td
Raising a structured array to a power will be returned as a structured array if possible:
https://wolfram.com/xid/0bdo09sse-lw5ui2
https://wolfram.com/xid/0bdo09sse-owi93v
https://wolfram.com/xid/0bdo09sse-wdjmwi
https://wolfram.com/xid/0bdo09sse-j45yx1
IdentityMatrix raised to any power is itself:
https://wolfram.com/xid/0bdo09sse-8vb729
More generally, the power of any diagonal matrix is the power of its diagonal elements:
https://wolfram.com/xid/0bdo09sse-7h79pn
Raise HilbertMatrix to a negative power:
https://wolfram.com/xid/0bdo09sse-gk8bid
Compute the power of a matrix of univariate polynomials of degree :
https://wolfram.com/xid/0bdo09sse-cupbp5
https://wolfram.com/xid/0bdo09sse-j0guy7
Applications (5)Sample problems that can be solved with this function
Find the fundamental solution for the constant coefficient system of difference equations :
https://wolfram.com/xid/0bdo09sse-i7hdrx
Define fundamental solution using MatrixPower:
https://wolfram.com/xid/0bdo09sse-bb4wx7
Show that it satisfies the equation:
https://wolfram.com/xid/0bdo09sse-sbvna
It satisfies the initial condition for a fundamental solution:
https://wolfram.com/xid/0bdo09sse-e9n3i5
Find the matrix exponential for a matrix without a full set of eigenvectors:
https://wolfram.com/xid/0bdo09sse-fgg4zt
https://wolfram.com/xid/0bdo09sse-cbrg65
Compute the exponential as the power series for each term:
https://wolfram.com/xid/0bdo09sse-y2eie9
https://wolfram.com/xid/0bdo09sse-b5b67u
Construct a rotation matrix as a limit of repeated infinitesimal transformations:
https://wolfram.com/xid/0bdo09sse-gu0
https://wolfram.com/xid/0bdo09sse-gky
Inverse power iteration for the smallest eigenvalue of a sparse positive definite matrix:
https://wolfram.com/xid/0bdo09sse-e9m8tc
https://wolfram.com/xid/0bdo09sse-k7qew0
https://wolfram.com/xid/0bdo09sse-bdprov
Shifted inverse power iteration for the largest eigenvalue:
https://wolfram.com/xid/0bdo09sse-h56h4r
https://wolfram.com/xid/0bdo09sse-j5za93
An easy way to evaluate a matrix polynomial:
https://wolfram.com/xid/0bdo09sse-bumnmg
https://wolfram.com/xid/0bdo09sse-b4tjud
Evaluate a characteristic polynomial:
https://wolfram.com/xid/0bdo09sse-dgf4i2
https://wolfram.com/xid/0bdo09sse-dhg5rx
https://wolfram.com/xid/0bdo09sse-m0a5y
Properties & Relations (10)Properties of the function, and connections to other functions
For a positive integer power , MatrixPower[m,n] is equivalent to ( times):
https://wolfram.com/xid/0bdo09sse-488g3
Write the formula more compactly with Apply (@@):
https://wolfram.com/xid/0bdo09sse-cexd40
For a negative integer power , MatrixPower[m,-n] is equivalent to ( times):
https://wolfram.com/xid/0bdo09sse-7uobuq
Write the formula more compactly with Apply:
https://wolfram.com/xid/0bdo09sse-rd2mf2
In particular, negative matrix powers are not defined for singular matrices:
https://wolfram.com/xid/0bdo09sse-ewl63b
For a nonsingular matrix m, MatrixPower[m,0] is the identity matrix:
https://wolfram.com/xid/0bdo09sse-i8qv5a
https://wolfram.com/xid/0bdo09sse-fu5ten
https://wolfram.com/xid/0bdo09sse-nlwvai
If m is nonsingular, MatrixPower[m, n].MatrixPower[m,-n] is the identity:
https://wolfram.com/xid/0bdo09sse-lguki5
https://wolfram.com/xid/0bdo09sse-hanknz
For noninteger powers, MatrixPower effectively uses the power series, with Power replaced by MatrixPower:
https://wolfram.com/xid/0bdo09sse-xbn62t
https://wolfram.com/xid/0bdo09sse-1hl7r6
Equivalently, MatrixPower is MatrixFunction applied to the appropriate function for the power:
https://wolfram.com/xid/0bdo09sse-dks9sn
The matrix power of a diagonal matrix is a diagonal matrix with the diagonal entries raised to that power:
https://wolfram.com/xid/0bdo09sse-enlw5a
https://wolfram.com/xid/0bdo09sse-dc9c5
https://wolfram.com/xid/0bdo09sse-bq7f2a
For any power and diagonalizable matrix , MatrixPower[m,s] equals :
https://wolfram.com/xid/0bdo09sse-w71q0i
https://wolfram.com/xid/0bdo09sse-n77187
Use JordanDecomposition to find a diagonalization:
https://wolfram.com/xid/0bdo09sse-1b5ari
https://wolfram.com/xid/0bdo09sse-e99dvg
https://wolfram.com/xid/0bdo09sse-udii8o
https://wolfram.com/xid/0bdo09sse-gsf7gz
For a real symmetric matrix s and integer power n, MatrixPower[s,n] is also real and symmetric:
https://wolfram.com/xid/0bdo09sse-jhd6et
The analogous statement is true for Hermitian matrices:
https://wolfram.com/xid/0bdo09sse-m9n2o3
For am orthogonal matrix o and any power s, MatrixPower[o,s] is also orthogonal:
https://wolfram.com/xid/0bdo09sse-ldg0ke
The analogous statement is true for unitary matrices:
https://wolfram.com/xid/0bdo09sse-8vduyj
can be computed from the JordanDecomposition as :
https://wolfram.com/xid/0bdo09sse-sgp8bx
Moreover, is zero except in upper-triangular blocks delineated by s in the superdiagonal:
https://wolfram.com/xid/0bdo09sse-o739h8
Wolfram Research (1991), MatrixPower, Wolfram Language function, https://reference.wolfram.com/language/ref/MatrixPower.html (updated 2024).
Text
Wolfram Research (1991), MatrixPower, Wolfram Language function, https://reference.wolfram.com/language/ref/MatrixPower.html (updated 2024).
Wolfram Research (1991), MatrixPower, Wolfram Language function, https://reference.wolfram.com/language/ref/MatrixPower.html (updated 2024).
CMS
Wolfram Language. 1991. "MatrixPower." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/MatrixPower.html.
Wolfram Language. 1991. "MatrixPower." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/MatrixPower.html.
APA
Wolfram Language. (1991). MatrixPower. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MatrixPower.html
Wolfram Language. (1991). MatrixPower. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MatrixPower.html
BibTeX
@misc{reference.wolfram_2024_matrixpower, author="Wolfram Research", title="{MatrixPower}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/MatrixPower.html}", note=[Accessed: 10-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_matrixpower, organization={Wolfram Research}, title={MatrixPower}, year={2024}, url={https://reference.wolfram.com/language/ref/MatrixPower.html}, note=[Accessed: 10-January-2025
]}