# Cayley-Hamilton theorem

Let be the set of complex -matrices and . Let

be the characteristic polynomial of , where is the identity matrix. The Cayley–Hamilton theorem says [a2], [a9] that every square matrix satisfies its own characteristic equation, i.e.

where is the zero-matrix.

The classical Cayley–Hamilton theorem can be extended to rectangle matrices. A matrix for may be written as , , . Let

Then the matrix () satisfies the equation [a8]

A matrix () may be written as

Let

Then the matrix () satisfies the equation [a8]

The Cayley–Hamilton theorem can be also extended to block matrices ([a4], [a13], [a15]). Let

(a1) |

where are commutative, i.e. for all . Let

be the matrix characteristic polynomial and let be the matrix (block) eigenvalue of , where denotes the Kronecker product. The matrix is obtained by developing the determinant of , considering its commuting blocks as elements [a15].

The block matrix (a1) satisfies the equation [a15]

Consider now a rectangular block matrix , where has the form (a1) and (). The matrix satisfies the equation [a4]

If , where has the form (a1) and , then

A pair of matrices is called regular if for some [a10], [a11], [a12]. The pair is called standard if there exist scalars such that . If the pair is regular, then the pair

(a2) |

is standard. If the pair is standard, then it is also commutative (). Let a pair be standard (commutative) and

Then the pair satisfies the equation [a1]

In a particular case, with , it follows that .

Let be the set of -order square complex matrices that commute in pairs and let be the set of square matrices partitioned in blocks belonging to .

Consider a standard pair of block matrices and let the matrix polynomial

be its matrix characteristic polynomial. The pair is called the block-eigenvalue pair of the pair .

Then [a6]

The Cayley–Hamilton theorem can be also extended to singular two-dimensional linear systems described by Roesser-type or Fomasini–Marchesini-type models [a3], [a14]. The singular two-dimensional Roesser model is given by

Here, is the set of non-negative integers; , respectively , are the horizontal, respectively vertical, semi-state vector at the point ; is the input vector; , () and () have dimensions compatible with and ; and

may be singular. The characteristic polynomial has the form

and the transition matrices , , are defined by

If , (the standard Roesser model), then the transition matrices may be computed recursively, using the formula , where ,

The matrices satisfy the equation [a3]

The singular two-dimensional Fornasini–Marchesini model is given by

where is the local semi-vector at the point , is the input vector, and is possibly singular. The characteristic polynomial has the form

and the transition matrices , , are defined by

The matrices satisfy the equation

The theorems may be also extended to two-dimensional continuous-discrete linear systems [a5].

#### References

[a1] | F.R. Chang, C.N. Chen, "The generalized Cayley–Hamilton theorem for standard pencils" Systems and Control Lett. , 18 (1992) pp. 179–182 |

[a2] | F.R. Gantmacher, "The theory of matrices" , 2 , Chelsea (1974) |

[a3] | T. Kaczorek, "Linear control systems" , I–II , Research Studies Press (1992/93) |

[a4] | T. Kaczorek, "An extension of the Cayley–Hamilton theorem for non-square blocks matrices and computation of the left and right inverses of matrices" Bull. Polon. Acad. Sci. Techn. , 43 : 1 (1995) pp. 49–56 |

[a5] | T. Kaczorek, "Extensions of the Cayley Hamilton theorem for -D continuous discrete linear systems" Appl. Math. and Comput. Sci. , 4 : 4 (1994) pp. 507–515 |

[a6] | T. Kaczorek, "An extension of the Cayley–Hamilton theorem for a standard pair of block matrices" Appl. Math. and Comput. Sci. , 8 : 3 (1998) pp. 511–516 |

[a7] | T. Kaczorek, "An extension of Cayley–Hamillon theorem for singular -D linear systems with non-square matrices" Bull. Polon. Acad. Sci. Techn. , 43 : 1 (1995) pp. 39–48 |

[a8] | T. Kaczorek, "Generalizations of the Cayley–Hamilton theorem for nonsquare matrices" Prace Sem. Podstaw Elektrotechnik. i Teor. Obwodów , XVIII–SPETO (1995) pp. 77–83 |

[a9] | P. Lancaster, "Theory of matrices" , Acad. Press (1969) |

[a10] | F.L. Lewis, "Cayley--Hamilton theorem and Fadeev's method for the matrix pencil " , Proc. 22nd IEEE Conf Decision Control (1982) pp. 1282–1288 |

[a11] | F.L. Lewis, "Further remarks on the Cayley–Hamilton theorem and Leverrie's method for the matrix pencil " IEEE Trans. Automat. Control , 31 (1986) pp. 869–870 |

[a12] | B.G. Mertzios, M.A. Christodoulous, "On the generalized Cayley–Hamilton theorem" IEEE Trans. Automat. Control , 31 (1986) pp. 156–157 |

[a13] | N.M. Smart, S. Barnett, "The algebra of matrices in -dimensional systems" Math. Control Inform. , 6 (1989) pp. 121–133 |

[a14] | N.J. Theodoru, "A Hamilton theorem" IEEE Trans. Automat. Control , AC–34 : 5 (1989) pp. 563–565 |

[a15] | J. Victoria, "A block-Cayley–Hamilton theorem" Bull. Math. Soc. Sci. Math. Roum. , 26 : 1 (1982) pp. 93–97 |

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Cayley-Hamilton theorem.

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