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https://texercises.com/exercise/trace-properties/
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Exercise:
abcliste abc forall A Bin textM_ntimes nKtextTrA BtextTrB A. abc If A and B are similar matrices then textTrAtextTrB. In particular if mathcalB and mathcalB' are two bases for V then textTrleftT_mathcalB^mathcalBrighttextTrleftT_mathcalB'^mathcalB'right. Thus textTrT is well defined. abcliste

Solution:
Proof. abcliste abc textTrA B_i^nA B_ii _i^n_j^n A_ij B_ji _ijA_ijB_ji _ijA_jiB_ij _i^n_j^n B_ijA_ji _i^nB A_ii textTrB A. abc If BP^-AP then textTrBtextTrP^-AP textTrP^- AP textTrAP P^- textTrA Regarding the claim about textTrT recall that T_mathcalB'^mathcalB'id_V_mathcalB'^mathcalB T_mathcalB^mathcalB id_V_mathcalB^mathcalB' and id_V_mathcalB'^mathcalBleftid_V_mathcalB^mathcalB'right^-. abcliste
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Exercise:
abcliste abc forall A Bin textM_ntimes nKtextTrA BtextTrB A. abc If A and B are similar matrices then textTrAtextTrB. In particular if mathcalB and mathcalB' are two bases for V then textTrleftT_mathcalB^mathcalBrighttextTrleftT_mathcalB'^mathcalB'right. Thus textTrT is well defined. abcliste

Solution:
Proof. abcliste abc textTrA B_i^nA B_ii _i^n_j^n A_ij B_ji _ijA_ijB_ji _ijA_jiB_ij _i^n_j^n B_ijA_ji _i^nB A_ii textTrB A. abc If BP^-AP then textTrBtextTrP^-AP textTrP^- AP textTrAP P^- textTrA Regarding the claim about textTrT recall that T_mathcalB'^mathcalB'id_V_mathcalB'^mathcalB T_mathcalB^mathcalB id_V_mathcalB^mathcalB' and id_V_mathcalB'^mathcalBleftid_V_mathcalB^mathcalB'right^-. abcliste
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characteristic polynomial, eth, fs23, lineare algebra, matrices, proof
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(3, default)
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ENG (English)
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