Construct a linear operator \(T\) with minimal polynomial \(x^2(x-1)^2\) and characteristic polynomial \(x^3(x-1)^4\). Describe the primary decomposition of the vector space under \(T\) and find the projections on the primary components. Find a basis in which the matrix of \(T\) is in Jordan form. Also find an explicit direct sum decomposition of the space into \(T\)-cyclic subspaces as in Theorem 3 and give the invariant factors.
Solution
Take \(V=F^7\) and, in the standard basis, let \[ T=J_2(0)\oplus J_1(0)\oplus J_2(1)\oplus J_2(1), \] where \(J_r(c)\) is the \(r\times r\) Jordan block with eigenvalue \(c\). Then \[ \chi_T(x)=x^3(x-1)^4, \qquad m_T(x)=x^2(x-1)^2, \] because the algebraic multiplicities are \(3\) and \(4\) and the largest block at each eigenvalue has size \(2\) [@HK71].
1. Primary decomposition and projections.
Let \[ W_0=\ker T^2, \qquad W_1=\ker(T-I)^2. \] For the displayed block matrix, \[ W_0=F^3\oplus0, \qquad W_1=0\oplus F^4, \qquad V=W_0\oplus W_1. \] The polynomials \[ e_0(x)=1-3x^2+2x^3, \qquad e_1(x)=3x^2-2x^3 \] satisfy \[ e_0\equiv1\pmod{x^2},\quad e_0\equiv0\pmod{(x-1)^2}, \] and the reverse congruences for \(e_1\). Hence the primary projections are \[ \boxed{E_0=e_0(T),\qquad E_1=e_1(T)}. \] They satisfy \(E_0+E_1=I\), \(E_i^2=E_i\), \(E_0E_1=0\), with ranges \(W_0,W_1\) respectively, by the primary-decomposition/CRT construction [@HK71].
2. Jordan basis.
The standard basis is already a Jordan basis for the displayed \(T\); its Jordan form is therefore \[ \boxed{J_2(0)\oplus J_1(0)\oplus J_2(1)\oplus J_2(1)}. \] This also directly verifies the characteristic and minimal polynomials above.
3. Cyclic decomposition and invariant factors.
Write the standard basis in the four displayed Jordan blocks as \((e_1,e_2)\), \((e_3)\), \((e_4,e_5)\), \((e_6,e_7)\), with the convention that the first vector of each size-two block is cyclic for that block. Define \[ \alpha_1=e_3+e_4, \qquad \alpha_2=e_1+e_6. \] Because the annihilators of \(e_3\) and \(e_4\) are respectively \(x\) and \((x-1)^2\), which are coprime, the annihilator of \(\alpha_1\) is \[ q_1=x(x-1)^2. \] Likewise the annihilators of \(e_1\) and \(e_6\) are \(x^2\) and \((x-1)^2\), so the annihilator of \(\alpha_2\) is \[ q_2=x^2(x-1)^2. \] Their cyclic subspaces use disjoint Jordan blocks, hence \[ V=Z(\alpha_1;T)\oplus Z(\alpha_2;T). \] Their dimensions are \(\deg q_1=3\) and \(\deg q_2=4\), totaling \(7\), and \(q_1\mid q_2\). Therefore the invariant factors are \[ \boxed{x(x-1)^2\mid x^2(x-1)^2}. \] Their product is \(x^3(x-1)^4\) and the last is the minimal polynomial, as required.