Cayley–Hamilton theorem

The Cayley - Hamilton (after Arthur Cayley and William Rowan Hamilton) is a set of linear algebra. He states that every square matrix is zero point of its characteristic polynomial.

Cayley - Hamilton

It is an n-dimensional K- vector space, and its characteristic polynomial. Then

This equation is to be understood as equality of pictures. In particular, is on the right side of the equation, the zero mapping and End ( V ) denotes the vector space of all linear maps from V to V.

So In particular, for each matrix

Conclusions

Simple conclusions from this set 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 lines.
• The inverse of an invertible matrix is smaller than a linear combination of powers of the matrix exponent as the number of lines displayed.
• The minimal polynomial of a matrix divides its characteristic polynomial.
• A square matrix with n -fold eigenvalue zero is nilpotent, since its characteristic polynomial is of the form.

In addition, this formula can be particularly simple formulas for higher powers of matrices find. For this purpose, the resulting polynomial is exempt with the matrices just after the required matrix.

Generalization

In the area of ​​commutative algebra, there are different unrelated generalizations of the theorem of Cayley - Hamilton for modules over commutative rings. In the following, such a generalization is given as an example.

Statement

There are a commutative ring with one element and a module which can be produced by elements. Next is an endomorphism of, for

Applies to an ideal. Then there is a monic polynomial with, so that is true.

Example

Let and and the ideal consisting of all even numbers. The endomorphism is defined by the matrix

Since all the coefficients of this matrix are just considered. The characteristic polynomial of is

Whose coefficients 2, -44 and 128 are, as claimed, multiples of 2, 4 or 8