On the Minimal Polynomial of a Vector

ABSTRACT

It is well known that the Cayley-Hamilton theorem is an interesting and important theorem in linear algebras, which was first explicitly stated by A. Cayley and W. R. Hamilton about in 1858, but the first general proof was published in 1878 by G. Frobenius, and numerous others have appeared since then, for example see [1,2]. From the structure theorem for finitely generated modules over a principal ideal domain it straightforwardly follows the Cayley-Hamilton theorem and the proposition that there exists a vector v in a finite dimensional linear space*V* such that and a linear transformation of *V* have the same minimal polynomial. In this note, we provide alternative proofs of these results by only utilizing the knowledge of linear algebras.

It is well known that the Cayley-Hamilton theorem is an interesting and important theorem in linear algebras, which was first explicitly stated by A. Cayley and W. R. Hamilton about in 1858, but the first general proof was published in 1878 by G. Frobenius, and numerous others have appeared since then, for example see [1,2]. From the structure theorem for finitely generated modules over a principal ideal domain it straightforwardly follows the Cayley-Hamilton theorem and the proposition that there exists a vector v in a finite dimensional linear space

Cite this paper

D. Zheng and H. Liu, "On the Minimal Polynomial of a Vector,"*Advances in Linear Algebra & Matrix Theory*, Vol. 2 No. 4, 2012, pp. 48-50. doi: 10.4236/alamt.2012.24008.

D. Zheng and H. Liu, "On the Minimal Polynomial of a Vector,"

References

[1] K. Hoffman and R. Kunze, “Linear Algebra,” 2nd Edition, Prentice Hall Inc., Upper Saddle River, 1971.

[2] R. A. Horn and C. R. Johnson, “Matrix Analysis,” Cambridge University Press, Cambridge, 1986.

[3] N. Jacobson, “Basic Algebra I,” 2nd Edition, W. H. Freeman and Company, New York, 1985.

[4] Th. W. Hungerford, “Algebra, GTM 73,” Springer-Verlag, New York, 1980. doi:10.1007/978-1-4612-6101-8

[1] K. Hoffman and R. Kunze, “Linear Algebra,” 2nd Edition, Prentice Hall Inc., Upper Saddle River, 1971.

[2] R. A. Horn and C. R. Johnson, “Matrix Analysis,” Cambridge University Press, Cambridge, 1986.

[3] N. Jacobson, “Basic Algebra I,” 2nd Edition, W. H. Freeman and Company, New York, 1985.

[4] Th. W. Hungerford, “Algebra, GTM 73,” Springer-Verlag, New York, 1980. doi:10.1007/978-1-4612-6101-8