Exercise:
abcliste abc Let Ain M_ntimes nK. Then P_AA. abc Let V be a finite dimensional vector space over K and Tin textEndV. Then P_TT. abcliste

Solution:
Proof. We first with Ain M_ntimes nmathbbC. bf Lemma . Suppose Ain M_ntimes nmathbbC is diagonalizable. Then P_AA. Proof. By asption Q^-AQ pmatrix lambda_ & & & ddots & & & lambda_n pmatrix. for some lambda_...lambda_nin mathbbC the eigenvalues of A. and some Qin textGLnmathbbC. We know that P_Alambda_...P_Alambda_n. Write P_Axc_nx^n+c_n-x^n-+...+c_x+c_. We have Q^-P_AAQc_nQ^-A^nQ+c_n-Q^-A^n-Q+...+c_Q^-AQ+c_Q^-IQ c_nQ^-AQ^n+c_n-Q^-AQ^n-+...+c_Q^-AQ+c_ I c_nLambda^n+c_n-Lambda^n-+...+c_Lambda+c_ I c_n pmatrix lambda_^n & & & ddots & & & lambda_n^n pmatrix+ c_n- pmatrix lambda_^n- & & & ddots & & & lambda_n^n- pmatrix+...+c_ pmatrix lambda_ & & & ddots & & & lambda_n pmatrix +c_ I pmatrix P_Alambda_ & & & ddots & & & P_Alambda_n pmatrix bf Lemma . Let Ain M_ntimes nmathbbC be any matrix. Then exists a sequence of matrices A_k kin mathbbZ_geq s.t. abcliste abc A_k is diagonalizable. abc lim_limitskrightarrow inftyA_kA. i.e. lim_limitskrightarrow inftyA_kijAij forall leq ileq n leq jleq n. abcliste Proof. We have seen that every matrix in M_ntimes nmathbbC is trigonalizable Longrightarrow exists Pin textGLn mathbbC s.t. P^-AP pmatrix lambda_ & & b_ij & ddots & & & lambda_n pmatrix where lambda_...lambda_nin mathbbC are the eigenvalues of A or more precisely the zeroes of P_Ax each of them appearing in the diagonal as many times as its algebraic multiplicity and b_ijin mathbbC leq i jleq n. Note that AP pmatrix lambda_ & & b_ij & ddots & & & lambda_n pmatrixP^-. Let Kin mathbbZ_geq . Pick lambda_k...lambda_nkin mathbbC s.t. abcliste abc |lambda_ik-lambda_i| frac k. abc lambda_k...lambda_nk are pairwise distinct. abclisteDefine A_k:P pmatrix lambda_k & & b_ij & ddots & & & lambda_nk pmatrixP^-. Clearly A_k has n distinct eigenvalues namely lambda_k...lambda_nk Longrightarrow A_k is diagonalizable alternatively P_A_kx splits o n distinct linear factors lambda_k-x... lambda_nk-x and forall i we have m_aA_k;lambda_ik We have lim_limitskrightarrow inftyA_klim_limitskrightarrow inftyP pmatrix lambda_k & & b_ij & ddots & & & lambda_nk pmatrixP^- P lim_limitskrightarrow infty pmatrix lambda_k & & b_ij & ddots & & & lambda_nk pmatrixP^- P pmatrix lambda_k & & b_ij & ddots & & & lambda_nk pmatrixP^- A forall given matrices PQin M_ntimes nK the map M_ntimes nmathbbClongrightarrow M_ntimes nmathbbC xlongmapsto Ptimes Q is continous. bf Lemma . Cayley-Hamilton holds for every matrix Ain M_ntimes nmathbbC. Proof. By Lemma exists A_k diagonalizable forall k s.t. A_klongrightarrow A. By Lemma P_A_kA_k forall k. But P_A_kA_klongrightarrow P_AA because P_BB deps continously on B every entry in P_BB is a polynomial function of the entries of B. Longrightarrow P_AA. bf Summary. We have proven Cayley-Hamilton over mathbbC. We'll see that this implies the theorem over every field K.