Celebratio Mathematica — Ranicki

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K. Y. Lam, A. Ranicki, and L. Smith: “Jordan normal form projections,” Arch. Math. 50 : 2 (1988), pp. 113–117. MR 930110 Zbl 0636.15005 article

Let \( K \) be a field, and let \( T:V \to V \) be an endomorphism of a finite-dimensional vector space \( V \) over \( K \) such that \( K \) contains the roots \( \lambda_1 \), \( \lambda_2,\dots \), \( \lambda_m \) of the characteristic polynomial \begin{align*} \chi(t) &= \det(tI - T)\\ &= (t - \lambda_1)^{n_1}(t - \lambda_2)^{n_2}\dots (t - \lambda_m)^{n_m}. \end{align*} \( V \) is a direct sum of \( T \)-invariant subspaces \[ V_j = \ker(T - \lambda_j I)^{n_j} ,\] one for each eigenvalue \( \lambda_j \), such that \[ T - \lambda_j I: V_k \to V_k \] is nilpotent for \( j = k \) and an automorphism for \( j \neq k \). We obtain in this paper an explicit formula for the projection \[ p_j(T) = p_j(T)^2: V \to V \] onto the subspace \( V_j \), as a polynomial in \( T \).

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Kee Yuen Lam	Related