Let \(A\in M_6(\QQ)\) have minimal polynomial \[ m_A=(x-1)(x^2+1)^2. \] Determine all possible rational canonical forms of \(A\).
Let \(B\in M_{10}(\QQ)\) have minimal polynomial \[ m_B=(x^2+1)^2(x^3+1). \] Determine all possible rational canonical forms of \(B\).
Solution
Recall that the invariant factors \[ d_1\mid d_2\mid\cdots\mid d_t \] satisfy \[ \chi_A=\prod_i d_i, \qquad m_A=d_t. \] Equivalently, one may list the elementary divisors and align the powers of each irreducible factor from the right.
1. The \(6\times6\) case is unique.Write\[ a=x-1, \qquad b=x^2+1. \]The minimal polynomial \(ab^2\) already contributes degree\[ 1+4=5. \]Since the characteristic polynomial has degree \(6\) and has the same irreducible factors as the minimal polynomial, the only possible additional elementary divisor has degree \(1\), namely another copy of \(a\).Thus the elementary divisors are\[ a,\ a,\ b^2. \]The invariant factors are therefore\[ d_1=a, \qquad d_2=ab^2. \]Hence there is exactly one rational canonical form:\[ \boxed{C(x-1)\oplus C\bigl((x-1)(x^2+1)^2\bigr)}. \]
2. The \(10\times10\) case has three possibilities.Factor\[ x^3+1=(x+1)(x^2-x+1). \]Put\[ b=x^2+1, \qquad c=x+1, \qquad d=x^2-x+1. \]Then\[ m_B=b^2cd \]has degree \(4+1+2=7\). We must add elementary divisors of total degree \(3\), without introducing any new irreducible factor or increasing an exponent beyond those in the minimal polynomial.The only possibilities are:
2.1. Add \(c\) and \(d\).The elementary divisors are\[ b^2,\ c,\ c,\ d,\ d. \]Hence\[ d_1=cd=x^3+1, \qquad d_2=b^2cd=m_B, \]and\[ \boxed{C(x^3+1)\oplus C(m_B)}. \]
2.2. Add \(c\) and \(b\).The elementary divisors are\[ b,\ b^2,\ c,\ c,\ d. \]Hence\[ d_1=bc=(x^2+1)(x+1), \qquad d_2=b^2cd=m_B, \]and\[ \boxed{C\bigl((x^2+1)(x+1)\bigr)\oplus C(m_B)}. \]
2.3. Add three further copies of \(c\).The elementary divisors are\[ b^2,\ c,\ c,\ c,\ c,\ d. \]The invariant factors are\[ d_1=c,\qquad d_2=c,\qquad d_3=c,\qquad d_4=b^2cd=m_B. \]Thus\[ \boxed{C(x+1)\oplus C(x+1)\oplus C(x+1)\oplus C(m_B)}. \]
These three exhaust the degree-\(3\) ways to augment the elementary divisors while preserving the given minimal polynomial.