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Let be the set of complex
-matrices and
. Let
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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.
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where is the zero-matrix.
The classical Cayley–Hamilton theorem can be extended to rectangle matrices. A matrix for
may be written as
,
,
. Let
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Then the matrix (
) satisfies the equation [a8]
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A matrix (
) may be written as
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Let
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Then the matrix (
) satisfies the equation [a8]
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The Cayley–Hamilton theorem can be also extended to block matrices ([a4], [a13], [a15]). Let
![]() | (a1) |
where are commutative, i.e.
for all
. Let
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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]
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Consider now a rectangular block matrix , where
has the form (a1) and
(
). The matrix
satisfies the equation [a4]
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If , where
has the form (a1) and
, then
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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
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Then the pair satisfies the equation [a1]
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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
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be its matrix characteristic polynomial. The pair is called the block-eigenvalue pair of the pair
.
Then [a6]
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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
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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
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may be singular. The characteristic polynomial has the form
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and the transition matrices ,
, are defined by
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If ,
(the standard Roesser model), then the transition matrices
may be computed recursively, using the formula
, where
,
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The matrices satisfy the equation [a3]
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The singular two-dimensional Fornasini–Marchesini model is given by
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where is the local semi-vector at the point
,
is the input vector,
and
is possibly singular. The characteristic polynomial has the form
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and the transition matrices ,
, are defined by
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The matrices satisfy the equation
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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 ![]() |
[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 ![]() |
[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 ![]() |
[a11] | F.L. Lewis, "Further remarks on the Cayley–Hamilton theorem and Leverrie's method for the matrix pencil ![]() |
[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 ![]() |
[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 |
Cayley-Hamilton theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cayley-Hamilton_theorem&oldid=11780