CAYLEY-HAMILTON THEOREM for SQUARE and RECTANGULAR MATRICES and BLOCK MATRICES

CAYLEY-HAMILTON THEOREM for SQUARE and RECTANGULAR MATRICES and BLOCK MATRICES

CAYLEY-HAMILTON THEOREM FOR SQUARE AND RECTANGULAR MATRICES AND BLOCK MATRICES

Paramjeet Sangwan

Assistant Professor, Geeta Engineering College, Naultha Panipat mathematicsat2012@gmail.com

Abstract: The main aim of the paper is to introduce cayley- Hamilton Theorem and also to explain its extension for the square and rectangular matrics. In this paper C-H Theorem extension for block matrics has also explained.

Keywords: Cayley- Hamilton Theorem, topology, Matrices, square, rectangular , block.

1. INTRODUCTION

   Defintion 1.1. If A is an n×n matrix ,then the characteristic polynomial of A is defined to be PA(x)= det(xI-A). This is a polynomial in x of degree n with leading term xn . the constant term c0 of a polynomial q(x) is interpreted as c0 I in q(A).

   Theorem 1.2 (Cayley Hamilton Theorem). If A is an n×n matrix ,then pA(A)=0, the zero matrix.

   Theorem 1.3 If q0 is a quaternion of the form q-= a+bi+cj+dk with a,b,c,d, being real , then q2 – 2aq + (a2

   2 2 2

   If one represents a quaternion q= a+bi+cj+dk as a matrix, A =[ a + bi c + di],

   c + di a bi

   PA(A) = A2 -2aA+(a2 + b2 + c2 + d2)I = 0, and the polynomial given in Theorem 1.3 is characteristic polynomial of A

2. GENERALIZATION OF CAYLEY HAMILTON

   THEOREM

   Theorem 2.1 (Cayley-Hamilton Theorem). For any n × n Matrix A, PA(A)=0.

   Proof. Let D(x) be the matrix with polynomial entries D(x)= adj(xIn-A), So D(x)(xI-A)= det(xIn-A)In. Since each entry in D(x) is the determinant of an (n-1) X( (n-1) submatrix of (xIn-A), each entry of D(x) is a polynomial of degree less than or equal to n-1. It folowws that there exist matrices D0, D1,Dn-1 with entries from C such that

   +b +c +d )=0

   q-1 = q

   |q|

   a-bi-cj-dk

   = a2+b2+c2+d2

   2a

   a+bi+cj+dk

   D(x) = Dn-1 xn-1++ D1 x +Do . Then the matrix equation follows

   det(xIn-A) In = (x In-A) adj(xIn-A) = (xIn-A)D(x)

   Substituting pA(x)= det(xIn-A), (and using the fact that scalars commute with matrix)

   XnIn+ bn-1xn-1In+.+ b1xIn+b0In

   = a2+b2+c2+d2 a2+b2+c2+d2

   1

   = a2+b2+c2+d2 (2a q)

   a2 + b2 + c2 + d2 = 2aq q2

   q2 2aq + (a2 + b2 + c2 + d2) = 0

   = pA(x) In= det(xIn-A)In

   =(xIn-A)adj(xIn-A)

   =(xIn-A)(xn-1Dn-1++xD1+Do)

   =xnDn-1- xn-1ADn-1+ xn-1Dn-2-xn-2ADn-2++ xD0-ADo

   =xnDn-1+ xn-1 (-ADn-1+ Dn-2 ) +..+(-AD1+Do) -ADo

   m am-i[An-i An-i-1A2] = 0mn (3.5)

   i=O 1 1

   Since two polynomials are equal if and only if their coefficients are equal , the coefficient matrices are equal ;

   Where 0mn is the (m×n) matrix. Theorem 3. Let ,

   that is , In=Dn-1, bn-1In= ( -ADn-1+Dn-2),,b1In=-

   (AD1+D0), and b0In=-AD0. This means that A may be

   A

   1

   A = A2 C

   m×n

   , m > n

   substituted for the variable x in the equation (2.1) to conclude

   PA(A) = An+ bn-1 An-1++ b1A+b0In

   and let the characteristic polynomial of A1 have the form . Then

   Am-i

   n n-1

   n i=O

   an-i

   1

   A2Am-i-1

   = 0mn

   (3.6)

   =A Dn-1+A

   (-ADn-1 +Dn-2)++A(-AD1+D0)-AD0 1

   =AnDn-1-AnDn-1+An-1Dn-2- An-1Dn-2+.AD0-AD0

   =0 This proves the theorem

3. CAYLEY-HAMILTON THEOREM FOR SQUARE AND RECTANGULAR MATRICES

   Let Cn×m be the set of complex (n×m) matrices.

   Theorem 1. (Cayley-Hamilton theorem). Let

   p(s) = det[Ins A]

4. CAYLEY-HAMILTON THEOREM FOR BLOCK MATRIX

   The classical Cayley-Hamilton theorem can be also extended for block matrices.

   Theorem 4. (Cayley-Hamilton theorem for block matrices).

   Let

   A11 A1m

   n A =

   Cmn×mn

   = L aisi (an = 1)

   i=O

   (3.1)

   Am1 Amm

   where Aij Cn×nare commutative i.e.,

   (4.1)

   be the characteristic polynomial of A, where In is the (n×n) identity matrix. Then

   Let

   AijAkl = AklAij for all i,j,k,l = 1,2,.,m

   (4.2)

   i=O

   p(A) = n

   aiAi = 0n (3.2)

   P(S) = det[Im A] = Sm

   + D1S

   m-1

   + +

   Where 0n is the (n×n) matrix.

   The classical Cayley-Hamilton theorem can be extended to rectangular matrices as follows [16]

   Theorem 2. (Cayley-Hamilton theorem for rectangular matrices).

   Let

   A = [A1 A2] Cm×n, A1 Cm×m, A2

   Dm-1S + Dm

   (4.3)

   be the matrix polynomial of A , where S Cn×n is the block matrix having eigenvalue of A, denotes the Kronecker product of matrix .

   Then

   Cm×(n-m), (n > m) (3.3)

   P(A) = m

   [Im

   Dm-i

   ]Ai = 0 (DO

   = In) (4.4)

   i=O

   and

   The matrix (4.3) is obtained by developing the determinant

   pA1

   = det[Ims A1] = m

   aisi (am = 1) (3.4)

   of the matrix [In

   S A] , considering its commuting

   i=O

   be the characteristic polynomial of A1. Then

   block as scalar entries.

   Theorem 5. (Cayley- Hamilton Theorem for rectangular block matrices)

   Let A = [A1 A2] Cmn×(mn+p) and let matrix

   charactristics polynomial of A have the form (4.3.2), then

   1. D.R. Wilkins, Linear Operators and the Cayley Hamilton Theorem , available from http://www.maths.tod.i.e/pub/histMath/people/Hamilton

      , 2005

   2. Raghib Abu Saris and Wajdi Ahmad, Avoiding Eigen values in Computing Matrix Powers. The

      m

      i=O

      [Im Dm-i][Ai+1 AiA2] = 0 (DO = In) (4.5)

      Mathematical Association of America 112(2005),

      Theorem 6. Let

      450-454.

   3. Kaczorek T., 1988. Vectors and Matrices in Automation and Electrotechnics, WNT Warszawa (in Polish).

      A = A C(mn+p)×mn , A C A2

      Cp×mn

      mn×mn

      , A2

   4. Kaczorek T., 1995. An existence of the Cayley-Hamilton

      theorem for nonsquare block matrices and computation of the left and right in verses of matrices, Bull. Pol. Acad.Techn. Sci., vol. 43, No 1, pp. 49-56.

      and let the matrix characteristic polynomial of A have the form , then

   5. Kaczorek T., 1998. An extension of the Cayley- Hamilton theorem for a standard pair of block matrices, Appl Math. And Com. Sci., vol. 8, No 3, pp. 511-516.

   m

   A

   L A [Im Dm-i]Ai = 0 (DO = In)

   2

   i=O

5. CHARACTERSTICS

   The Cayley Hamilton theorem is one of the most powerful and classical matrix theory theorem. Many application derive their results from this theorem. To understand the scope of this theorem , alternate proofs were used. Each proof helped to understand how intertwined areas of mathematics are with respect to matrices and the characteristics polynomial.

6. APPLICATION OF CAYLEY HAMILTON

THEOREM

A very common application of the Cayley- Hamilton Theorem is to use it to find An usually for the large powers of n. However many of the techniques involved require the use of the eigen values of A.

REFERENCES

1. William A. Adkins and Mark G. Davidson, The Cayley Hamilton and Frobenious theorems via the Laplace Transformation, Linear Algebra and its Applications 371(2003), 147-152.

2. Arthur Cayley, A memoir on the theory of Matrices, available from http:// www.jstor.org, 1857.

3. Wikipedia , Arthur Cayley, Available from http:// en.wikipedia .org, 2004

4. Wikipedia, William Rowan Hamilton, Available from http:// en. wikipedia .org, 2005