A Generalization of the Cayley-Hamilton Theorem

Affiliation(s)

Department of Mathematics and Statistics, Himachal Pradesh University, Shimla, India.

Department of Mathematics and Statistics, Himachal Pradesh University, Shimla, India.

ABSTRACT

It is proposed to generalize the concept of the famous classical Cayley-Hamilton theorem for square matrices wherein for any square matrix*A*, the det (*A-xI*) is replaced by det *f*(*x*) for arbitrary polynomial matrix *f*(*x*).

It is proposed to generalize the concept of the famous classical Cayley-Hamilton theorem for square matrices wherein for any square matrix

Cite this paper

R. Kanwar, "A Generalization of the Cayley-Hamilton Theorem,"*Advances in Pure Mathematics*, Vol. 3 No. 1, 2013, pp. 109-115. doi: 10.4236/apm.2013.31014.

R. Kanwar, "A Generalization of the Cayley-Hamilton Theorem,"

References

[1] J. Gilbert and L. Gilbert, “Linear Algebra and Matrix Theory,” Academic Press Inc., New York, 1995.

[2] K. Hoffman and R. Kunze, “Linear Algebra,” Prentice Hall of India, New Delhi, 2010.

[3] P. Lancaster, “Theory of Matrices,” Academic Press, New York, 1969.

[4] F. R. Gantmatcher, “Theory of Matrices, Vol. 2,” Chelsea Publishing, New York, 1974.

[5] T. Kaczorek, “An Existence of the Cayley-Hamilton Theorem for Singular 2-D Linear Systems with Non-Square Matrices,” Bulletin of the Polish Academy of Sciences. Technical Sciences, Vol. 43, No. 1, 1995, pp. 39-48.

[6] T. Kaczorek, “Generalization of the Cayley-Hamilton Theorem for Non-Square Matrices,” International Conference of Fundamentals of Electronics and Circuit Theory XVIII- SPETO, Gliwice, 1995, pp. 77-83.

[7] T. Kaczorek, “An Existence of the Caley-Hamilton Theorem for Non-Square Block Matrices,” Bulletin of the Polish Academy of Sciences. Technical Sciences, Vol. 43, No. 1, 1995, pp. 49-56.

[8] T. Kaczorek, “An Extension of the Cayley-Hamilton Theorem for a Standard Pair of Block Matrices,” Applied Mathematics and Computation Sciences, Vol. 8, No. 3, 1998, pp. 511-516.

[9] F. R. Chang and C. N. Chan, “The Generalized Cayley-Hamilton Theorem for Standard Pencils,” Systems & Control Letters, Vol. 18, No. 3, 1992, pp. 179-182. doi:10.1016/0167-6911(92)90003-B

[10] F. L. Lewis, “Cayley-Hamilton Theorem and Fadeev’s Method for the Matrix Pencil [sE-A],” 22nd IEEE Conference on Decision and Control, San Diego, 1982, pp. 1282-1288.

[11] F. L. Lewis, “Further Remarks on the Cayley-Hamilton Theorem and Fadeev’s Method for the Matrix Pencil [sE-A],” IEEE Transactions on Automatic Control, Vol. 31, No. 7, 1986. pp. 869-870. doi:10.1109/TAC.1986.1104420

[12] T. Kaczorek, “Extensions of the Cayley-Hamilton Theorem for 2D Continuous-Discrete Linear Systems,” Applied Mathematics and Computation Sciences, Vol. 4, No. 4, 1994, pp. 507-515.

[13] N. M. Smart and S. Brunett, “The Algebra of Matrices in n-Dimensional Systems,” IMA Journal of Mathematical Control and Information, Vol. 6, No. 2, 1989, pp. 121-133. doi:10.1093/imamci/6.2.121

[14] M. Buslowicz and T. Kaczorek, “Reachability and Minimum Energy Control of Positive Linear Discrete-Time Systems with One Delay,” Proceedings of 12th Mediterranean Conference on Control and Automation, Kasadesi- Izmur, CD ROM, 2004.

[15] T. Kaczorek, “Linear Control Systems, Vol. I and II,” Research Studies Press, Taunton, 1992-1993.

[16] B. G. Mcrtizios and M. A. Christodolous, “On the Generalized Cayley-Hamilton Theorem,” IEEE Transactions on Automatic Control, Vol. 31, No. 1, 1986, pp. 156-157.

[17] N. J. Theodoru, “M-Dimensional Cayley-Hamilton Theorem,” IEEE Transactions on Automatic Control, Vol. 34, No. 5, 1989, pp. 563-565. doi:10.1109/9.24217

[18] M. Buslowicz, “An Algorithm of Determination of the Quasi-Polynomial of Multivariate Time-Invariant Linear System with Delays Based on State Equations,” Archive of Automatics and Telemechanics, Vol. 36, No. 1, 1981, pp. 125-131.

[19] M. Buslowicz, “Inversion of Characteristic Matrix of the Time-Delay Systems of Neural Type,” Control Engineering, Vol. 7, No. 4, 1982, pp. 195-210.

[20] T. Kaczorek, “Extension of the Cayley-Hamilton Theorem for Continuous-Time Systems with Delays,” International Journal of Applied Mathematics and Computer Science, Vol. 15, No. 2, 2005, pp. 231-234.

[21] T. Kaczorek, “Vectors and Matrices in Automation and Electrotechnics,” Polish Scientific Publishers, Warsaw, 1988.

[1] J. Gilbert and L. Gilbert, “Linear Algebra and Matrix Theory,” Academic Press Inc., New York, 1995.

[2] K. Hoffman and R. Kunze, “Linear Algebra,” Prentice Hall of India, New Delhi, 2010.

[3] P. Lancaster, “Theory of Matrices,” Academic Press, New York, 1969.

[4] F. R. Gantmatcher, “Theory of Matrices, Vol. 2,” Chelsea Publishing, New York, 1974.

[5] T. Kaczorek, “An Existence of the Cayley-Hamilton Theorem for Singular 2-D Linear Systems with Non-Square Matrices,” Bulletin of the Polish Academy of Sciences. Technical Sciences, Vol. 43, No. 1, 1995, pp. 39-48.

[6] T. Kaczorek, “Generalization of the Cayley-Hamilton Theorem for Non-Square Matrices,” International Conference of Fundamentals of Electronics and Circuit Theory XVIII- SPETO, Gliwice, 1995, pp. 77-83.

[7] T. Kaczorek, “An Existence of the Caley-Hamilton Theorem for Non-Square Block Matrices,” Bulletin of the Polish Academy of Sciences. Technical Sciences, Vol. 43, No. 1, 1995, pp. 49-56.

[8] T. Kaczorek, “An Extension of the Cayley-Hamilton Theorem for a Standard Pair of Block Matrices,” Applied Mathematics and Computation Sciences, Vol. 8, No. 3, 1998, pp. 511-516.

[9] F. R. Chang and C. N. Chan, “The Generalized Cayley-Hamilton Theorem for Standard Pencils,” Systems & Control Letters, Vol. 18, No. 3, 1992, pp. 179-182. doi:10.1016/0167-6911(92)90003-B

[10] F. L. Lewis, “Cayley-Hamilton Theorem and Fadeev’s Method for the Matrix Pencil [sE-A],” 22nd IEEE Conference on Decision and Control, San Diego, 1982, pp. 1282-1288.

[11] F. L. Lewis, “Further Remarks on the Cayley-Hamilton Theorem and Fadeev’s Method for the Matrix Pencil [sE-A],” IEEE Transactions on Automatic Control, Vol. 31, No. 7, 1986. pp. 869-870. doi:10.1109/TAC.1986.1104420

[12] T. Kaczorek, “Extensions of the Cayley-Hamilton Theorem for 2D Continuous-Discrete Linear Systems,” Applied Mathematics and Computation Sciences, Vol. 4, No. 4, 1994, pp. 507-515.

[13] N. M. Smart and S. Brunett, “The Algebra of Matrices in n-Dimensional Systems,” IMA Journal of Mathematical Control and Information, Vol. 6, No. 2, 1989, pp. 121-133. doi:10.1093/imamci/6.2.121

[14] M. Buslowicz and T. Kaczorek, “Reachability and Minimum Energy Control of Positive Linear Discrete-Time Systems with One Delay,” Proceedings of 12th Mediterranean Conference on Control and Automation, Kasadesi- Izmur, CD ROM, 2004.

[15] T. Kaczorek, “Linear Control Systems, Vol. I and II,” Research Studies Press, Taunton, 1992-1993.

[16] B. G. Mcrtizios and M. A. Christodolous, “On the Generalized Cayley-Hamilton Theorem,” IEEE Transactions on Automatic Control, Vol. 31, No. 1, 1986, pp. 156-157.

[17] N. J. Theodoru, “M-Dimensional Cayley-Hamilton Theorem,” IEEE Transactions on Automatic Control, Vol. 34, No. 5, 1989, pp. 563-565. doi:10.1109/9.24217

[18] M. Buslowicz, “An Algorithm of Determination of the Quasi-Polynomial of Multivariate Time-Invariant Linear System with Delays Based on State Equations,” Archive of Automatics and Telemechanics, Vol. 36, No. 1, 1981, pp. 125-131.

[19] M. Buslowicz, “Inversion of Characteristic Matrix of the Time-Delay Systems of Neural Type,” Control Engineering, Vol. 7, No. 4, 1982, pp. 195-210.

[20] T. Kaczorek, “Extension of the Cayley-Hamilton Theorem for Continuous-Time Systems with Delays,” International Journal of Applied Mathematics and Computer Science, Vol. 15, No. 2, 2005, pp. 231-234.

[21] T. Kaczorek, “Vectors and Matrices in Automation and Electrotechnics,” Polish Scientific Publishers, Warsaw, 1988.