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Exercise:
m_gT;lambdaleq m_aT;lambda

Solution:
Proof. Let v_...v_k be a basis for textEig_Tlambda. We ext this basis to a basis mathcalBv_...v_kw_k+...w_n f V. Then we have T_mathcalB^mathcalB pmatrix vdots & & vdots & vdots & & vdots Tv__mathcalB & hdots & Tv_k_mathcalB & Tw_k+_mathcalB & hdots & Tw_n_mathcalB vdots & & vdots & vdots & & vdots pmatrix. leftarray@c|c@ matrix lambda & & &ddots & & & lambda matrix & * hline matrix matrix & C arrayright textwhere Cin M_n-ktimes n-kK Longrightarrow P_TxtextdetleftT_mathcalB^mathcalB-x I_n... lambda-x^n textdetC-xI_n-k lambda-x^k P_Cx -^kx-lambda^k P_Cx. &Longrightarrow textthe multiplicity of lambda as a zero of P_Tx is at least k &Longrightarrow m_aT;lambdageq k m_gT;lambda.
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Exercise:
m_gT;lambdaleq m_aT;lambda

Solution:
Proof. Let v_...v_k be a basis for textEig_Tlambda. We ext this basis to a basis mathcalBv_...v_kw_k+...w_n f V. Then we have T_mathcalB^mathcalB pmatrix vdots & & vdots & vdots & & vdots Tv__mathcalB & hdots & Tv_k_mathcalB & Tw_k+_mathcalB & hdots & Tw_n_mathcalB vdots & & vdots & vdots & & vdots pmatrix. leftarray@c|c@ matrix lambda & & &ddots & & & lambda matrix & * hline matrix matrix & C arrayright textwhere Cin M_n-ktimes n-kK Longrightarrow P_TxtextdetleftT_mathcalB^mathcalB-x I_n... lambda-x^n textdetC-xI_n-k lambda-x^k P_Cx -^kx-lambda^k P_Cx. &Longrightarrow textthe multiplicity of lambda as a zero of P_Tx is at least k &Longrightarrow m_aT;lambdageq k m_gT;lambda.
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eigenvalue, eigenvector, eth, fs23, lineare algebra, proof
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(3, default)
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0 (default)
Language
ENG (English)
Type
Proof
Creator rk
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