There is an invertible matrix
such that

where is the companion Matrix for any Monic Polynomial

with . The Polynomials are called the ``invariant factors'' of , and satisfy for , ..., 1 (Hartwig 1996).

**References**

Gantmacher, F. R. *The Theory of Matrices, Vol. 1.* New York: Chelsea, 1960.

Hartwig, R. E. ``Roth's Removal Rule and the Rational Canonical Form.'' *Amer. Math. Monthly* **103**, 332-335, 1996.

Herstein, I. N. *Topics in Algebra, 2nd ed.* New York: Springer-Verlag, p. 162, 1975.

Hoffman, K. and Kunze, K. *Linear Algebra, 3rd ed.* Englewood Cliffs, NJ: Prentice-Hall, 1996.

Lancaster, P. and Tismenetsky, M. *The Theory of Matrices, 2nd ed.* New York: Academic Press, 1985.

Turnbull, H. W. and Aitken, A. C. *An Introduction to the Theory of Canonical Matrices, 2nd impression.*
New York: Blackie and Sons, 1945.

© 1996-9

1999-05-25