Rational Canonical Form

There is an invertible matrix ${\hbox{\sf Q}}$ such that

$${\hbox{\sf Q}}^{-1}{\hbox{\sf T}}{\hbox{\sf Q}}=\mathop{\rm diag}[L(\psi_1), L(\psi_2), \ldots, L(\psi_s)],$$

where $L(f)$ is the companion Matrix for any Monic Polynomial

$$f(\lambda)=f_0+f_1\lambda+\ldots+f_n\lambda^n$$

with $f_n=1$. The Polynomials $\psi_i$ are called the ``invariant factors'' of ${\hbox{\sf T}}$, and satisfy $\psi_{i+1}\vert\psi_i$ for $i=s-1$, ..., 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.