Cayley-Hamilton theorem

The Cayley-Hamilton theorem (after Arthur Cayley and William Rowan Hamilton ) is a set of the linear algebra . It says that every square matrix is ​​the zero of its characteristic polynomial .

background

Be a body , for example the body of real numbers or the body of complex numbers . For a given natural number , the square matrices with entries from each other can be linked by the arithmetic operations matrix addition , matrix multiplication and scalar multiplication (element-wise multiplication with elements of the field ). In these calculations, these matrices form an associative and unitary algebra over , with the identity matrix as one element . ${\displaystyle K}$ ${\displaystyle n}$ ${\displaystyle n\times n}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle K}$

Let be a vector space over a body with dimension . By choosing a base , the endomorphisms of (the linear mappings from to ) can be identified with the quadratic matrices with entries . The endomorphisms are mapped onto the respective mapping matrices. The multiplication of two matrices corresponds to the execution of the corresponding endomorphisms one after the other , the unit matrix corresponds to the identical mapping , and the endomorphisms of thus also form an associative and unitary algebra like the mapping matrices. ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle n}$ ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle n\times n}$ ${\displaystyle K}$ ${\displaystyle V}$

For a body denotes the ring of polynomials with coefficients from and the variables . Any such polynomial ${\displaystyle K}$ ${\displaystyle K[t]}$ ${\displaystyle K}$ ${\displaystyle t}$

${\displaystyle p(t)=a_{0}\cdot t^{0}+a_{1}\cdot t^{1}+a_{2}\cdot t^{2}+\ldots +a_{m-1}\cdot t^{m-t}+a_{m}\cdot t^{m}}$

With

${\displaystyle a_{0},a_{1},a_{2},\dots ,a_{m-1},a_{m}\in K}$

Defined for a given unitary associative algebra by mapping the algebra into itself by inserting a given algebra element into the polynomial and then replacing the operations that appear in the polynomial with the corresponding operations of the algebra, ${\displaystyle K}$ ${\displaystyle A}$

${\displaystyle p(A)=a_{0}\cdot A^{0}+a_{1}\cdot A^{1}+a_{2}\cdot A^{2}+\dots +a_{m-1}\cdot A^{m-1}+a_{m}\cdot A^{m}\,.}$

In the case of the algebra of matrices with entries from this in particular in appearing powers of by appropriate matrix powers of replaced, with equal -Einheitsmatrix. ${\displaystyle n\times n}$ ${\displaystyle K}$ ${\displaystyle p(t)}$ ${\displaystyle t}$ ${\displaystyle A}$ ${\displaystyle A^{0}}$ ${\displaystyle n\times n}$

Cayley-Hamilton theorem

Let be a finite-dimensional vector space over a body , let be an endomorphism of , and let be the characteristic polynomial of . Cayley-Hamilton's theorem now says that (that is, applied to itself) the zero mapping is on , i.e. H. the linear mapping that maps all elements of to the zero vector of . ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle F}$ ${\displaystyle V}$ ${\displaystyle P_{F}\in K[t]}$ ${\displaystyle F}$ ${\displaystyle P_{F}\left(F\right)}$ ${\displaystyle P_{F}}$ ${\displaystyle F}$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle V}$

In particular holds for every matrix ${\displaystyle A\in K^{n\times n}}$

${\displaystyle P_{A}(A)=0}$.

Here the zero matrix is in . ${\displaystyle 0}$ ${\displaystyle K^{n\times n}}$

In summary it can be said: every square matrix satisfies its characteristic equation.

Inferences

Simple implications from this theorem are:

• The powers of a square matrix span a subspace of the vector space of all square matrices, which has at most the dimension of the number of rows . ${\displaystyle n}$
• The inverse of an invertible matrix can be represented as a linear combination of the powers of the matrix with exponents smaller than the number of rows.
• The minimal polynomial of a matrix divides its characteristic polynomial .
• A square matrix with n-fold eigenvalue zero is nilpotent because its characteristic polynomial is of the form . ${\displaystyle \lambda ^{n}}$

In addition, this formula can be used to find particularly simple formulas for higher potencies of matrices. To do this, the resulting polynomial with the matrices is simply to be freed for the matrix you are looking for.

generalization

In the area of commutative algebra , there are several related generalizations of the Cayley-Hamilton theorem for modules about commutative rings . Such a generalization is given below with an example.

statement

Let there be a commutative ring with one element and one module that can be generated by elements. Further be an endomorphism of , for ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle n}$ ${\displaystyle f}$ ${\displaystyle M}$

${\displaystyle f(M)\subseteq IM}$

holds for an ideal . Then there is a normalized polynomial with such that . ${\displaystyle I\subseteq R}$ ${\displaystyle p(X)=X^{n}+a_{1}X^{n-1}+\cdots +a_{n}}$ ${\displaystyle a_{i}\in I^{i}}$ ${\displaystyle p\left(f\right)=0}$

example

Let and and the ideal consisting of all even numbers. Let the endomorphism be defined by the matrix ${\displaystyle R=\mathbb {Z} }$ ${\displaystyle M=\mathbb {Z} ^{3}}$ ${\displaystyle I=2\mathbb {Z} }$ ${\displaystyle f}$

${\displaystyle A={\begin{pmatrix}2&-2&6\\-2&-6&2\\4&-2&2\end{pmatrix}}}$.

Since all the coefficients of this matrix are even, the following applies . The characteristic polynomial of is ${\displaystyle f(M)\subseteq 2\mathbb {Z} \cdot M}$ ${\displaystyle f}$

${\displaystyle P_{f}(t)=t^{3}+2t^{2}-44t-128}$.

Its coefficients are 2, -44 and -128, as claimed, multiples of 2, 4 and 8, respectively.

Web links

Wikiversity: proof of the theorem on Wikiversity  course materials

• Proof of the sentence in the evidence archive

source

• Gerd Fischer: Linear Algebra . 14th, through Edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03217-0 .

Individual evidence

1. ↑ Hansjörg Dirschmid: Mathematical Foundations of Electrical Engineering . 4th edition. Springer-Verlag, 1990, ISBN 3-322-83228-7 , p. 545 .
2. ↑ Wolmer V. Vasconcelos: Integral closure . Springer, Berlin 2005, ISBN 3-540-25540-0 , pp. 66 ff .
3. ^ David Eisenbud: Commutative algebra with view toward algebraic geometry . Springer, New York 1997, ISBN 3-540-94269-6 , pp. 120 .