Matrix potency

In linear algebra , the matrix power is the result of a repeated matrix multiplication .

definition

The power of a square matrix over a half ring is defined as repeated multiplication , analogous to the power of a number. If is a square matrix, then is ${\displaystyle A\in R^{n\times n}}$ ${\displaystyle R}$ ${\displaystyle A}$

${\displaystyle A^{0}=E;\quad A^{1}=A;\quad A^{2}=A\cdot A;\quad A^{3}=A\cdot A\cdot A}$ etc.

General:

${\displaystyle A^{n}=\underbrace {A\cdot A\dotsc A} _{n{\text{-mal}}}}$.

The power is more formally defined recursively : If a square matrix, then is ${\displaystyle A}$

• ${\displaystyle A^{0}=E}$ and
• for all true . ${\displaystyle k\in \mathbb {N} =\{0,1,2,\dotsc \}}$ ${\displaystyle A^{k+1}=A^{k}\cdot A}$

properties

The power laws apply : applies to all ${\displaystyle n,m\in \mathbb {N} }$

• ${\displaystyle A^{n+m}=A^{n}\cdot A^{m}}$,
• ${\displaystyle A^{n\cdot m}=\left(A^{n}\right)^{m}}$.

Generalizations

Negative exponents

Powers with negative integer exponents are also defined for invertible matrices . The notation for the inverse matrix can also be interpreted as a matrix power. For negative exponents , you just put ${\displaystyle A^{-1}}$ ${\displaystyle -n}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle A^{-n}=\left(A^{-1}\right)^{n}}$.

Fractional exponents

Matrix powers with non-integer exponents, for example the square root of a matrix , can only be defined in special cases.

In some cases the matrix power can be reduced to the power of real numbers . If the matrix can be diagonalized , i.e. if a regular matrix and a diagonal matrix with (i.e. is similar to ) exist , then the following applies ${\displaystyle A}$ ${\displaystyle T}$ ${\displaystyle D}$ ${\displaystyle A=T\cdot D\cdot T^{-1}}$ ${\displaystyle A}$ ${\displaystyle D}$

${\displaystyle A^{n}=T\cdot D^{n}\cdot T^{-1}\ .}$

The power of a diagonal matrix is ​​obtained by raising the diagonal elements to the power. If the diagonal elements of (i.e. the eigenvalues of ) are positive, the above power laws also remain valid for fractional exponents. ${\displaystyle D}$ ${\displaystyle A}$

If a matrix cannot be diagonalized, one finds a useful generalization of the matrix power over the binomial series . A quick calculation method for this generalization can be obtained from the Jordan normal form . If the Jordan is decomposed, then applies ${\displaystyle A=T\cdot J\cdot T^{-1}}$

${\displaystyle A^{n}=T\cdot J^{n}\cdot T^{-1}}$

Efficient calculation

If the exponent is an integer , the matrix power can be calculated efficiently using binary exponentiation . The restrictions on the number range of the matrix elements are slight:

• If the exponent is non-negative, the matrix elements must be in a ring.
• If the exponent is negative, the matrix elements must be in a body.

If the number range of the matrix elements is algebraically closed , so any algebraic equations can be solved in it, the exponent can also be rational and the matrix power can be reduced to powers of scalar values using the Jordan normal form of , see above . ${\displaystyle A}$

Applications

Polynomials and power series

Using the matrix power, polynomials can also be defined for matrices. An example of this is e.g. B. the minimal polynomial . In the same way one can define power series for matrices, the most important series are the matrix logarithm , the matrix exponential and the Neumann series .

Graph theory

By suitable choice of the underlying half-ring , finding the shortest paths in a graph can be traced back to the calculation of a power of the adjacency matrix of the graph. The min-plus matrix multiplication is obtained by choosing the extended real numbers as the carrier set of . The addition in then corresponds to the formation of the minimum in and the multiplication in the addition in , whereby one sets. The absorbing zero in is then , while the one element in in is represented by the number . Is now the cost matrix of a graph with nodes , then the associated distance matrix with the lengths of the shortest paths between all nodes of the graph. Since the addition is idempotent in is . ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle \mathbb {R} ^{\ast }=\mathbb {R} ^{+}\cup \{\infty \}}$ ${\displaystyle R}$ ${\displaystyle \mathbb {R} ^{\ast }}$ ${\displaystyle R}$ ${\displaystyle \mathbb {R} ^{\ast }}$ ${\displaystyle x+\infty =\infty +x=\infty }$ ${\displaystyle R}$ ${\displaystyle \infty }$ ${\displaystyle R}$ ${\displaystyle 0}$ ${\displaystyle K\in R^{n\times n}}$ ${\displaystyle n}$ ${\displaystyle D=\sum _{k=0}^{n}K^{k}}$ ${\displaystyle R}$ ${\displaystyle D=(1+K)^{n}}$

Other uses

• In theoretical economics and biology , matrix potencies are used to analyze long-term population developments, for example using a Leslie matrix .
• There are also applications for broadening the stereo base .

literature

• Peter Knabner , Wolf Barth : Lineare Algebra. Basics and applications (=  Springer textbook ). Springer Spectrum, Berlin et al. 2013, ISBN 978-3-642-32185-6 .
• Gilbert Strang : Linear Algebra . An introduction for first-year students (=  Springer textbook ). Springer, Berlin 2003, ISBN 3-540-43949-8 (English: Introduction to linear algebra . Translated by Michael Dellnitz).

Individual evidence

1. ↑ population development. (PDF; 72 kB) Retrieved September 6, 2013 .