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Exercise:
Let Ain M_ntimes nK. Then: abcliste abc P_Ax is a polynomial of degree n with leading coefficient -^n. i.e. P_Axc_nx^n+c_n-x^n-+...+c_x+c_ with c_iin Kquad forall i and c_n-^n. abc P_A^TxP_Ax. abc If A pmatrix lambda_ & hdots & * vdots & lambda_i & vdots & hdots & lambda_n pmatrix. has upper triangular form then P_Axlambda_-x ...lambda_n-x. abc If A leftarray@c|c@ matrix B_ matrix & * hline matrix matrix & B_ arrayright is a block matrix then P_AxP_B_x P_B_x. abc If A and B are similar matrices i.e. BP^-AP for some invertible matrix P then P_AxP_Bx. abcliste

Solution:
abcliste abc P_AxtextdetA-x I textdet pmatrix A_-x & A_& hdots & A_n A_ & A_-x& hdots & A_n vdots & vdots& vdots & vdots A_n & A_n& hdots & A_nn-x pmatrix. _sigmain S_ntextsgnsigma B_sigma_ ... B_nsigma_n A_-x A_-x A_nn-x+textproducts of degree leq n- -^nx^n+textterms of degree leq n- abc P_A^TxtextdetA^T-x I textdetleftA-x I^Tright textdetA-x I P_Ax abc todo exercise abc todo exercise abc P_BxtextdetB-x I textdetP^-AP-x I textdetP^-A-x IP textdetA-x I P_Ax. abcliste
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Exercise:
Let Ain M_ntimes nK. Then: abcliste abc P_Ax is a polynomial of degree n with leading coefficient -^n. i.e. P_Axc_nx^n+c_n-x^n-+...+c_x+c_ with c_iin Kquad forall i and c_n-^n. abc P_A^TxP_Ax. abc If A pmatrix lambda_ & hdots & * vdots & lambda_i & vdots & hdots & lambda_n pmatrix. has upper triangular form then P_Axlambda_-x ...lambda_n-x. abc If A leftarray@c|c@ matrix B_ matrix & * hline matrix matrix & B_ arrayright is a block matrix then P_AxP_B_x P_B_x. abc If A and B are similar matrices i.e. BP^-AP for some invertible matrix P then P_AxP_Bx. abcliste

Solution:
abcliste abc P_AxtextdetA-x I textdet pmatrix A_-x & A_& hdots & A_n A_ & A_-x& hdots & A_n vdots & vdots& vdots & vdots A_n & A_n& hdots & A_nn-x pmatrix. _sigmain S_ntextsgnsigma B_sigma_ ... B_nsigma_n A_-x A_-x A_nn-x+textproducts of degree leq n- -^nx^n+textterms of degree leq n- abc P_A^TxtextdetA^T-x I textdetleftA-x I^Tright textdetA-x I P_Ax abc todo exercise abc todo exercise abc P_BxtextdetB-x I textdetP^-AP-x I textdetP^-A-x IP textdetA-x I P_Ax. abcliste
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Tags
characteristic polynomial, eigenvalue, eigenvector, eth, fs23, lineare algebra, proof
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Difficulty
(3, default)
Points
0 (default)
Language
ENG (English)
Type
Proof
Creator rk
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