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 |
