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https://texercises.com/exercise/diagonalizable-endomorphisms-and-eigenvalues/
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Exercise:
Let Ain M_ntimes nK or Tin textEndV be diagonalizable. Then P_Ax decomposes as a product of linear i.e. deg factors P_Axlambda_-x...lambda_n-x where lambda_...lambda_n are the eigenvalues of A but there might be repetitions among the lambda_i's

Solution:
Proof. By astion exists an invertible matrix P s.t. P^-AP pmatrix lambda_ & & & ddots & & & lambda_n pmatrixtext denoted by Lambda. Longrightarrow P_AxP_Lambdaxtextdet pmatrix lambda_-x & & & ddots & & & lambda_n-x pmatrix lambda_-x...lambda_n-x
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Exercise:
Let Ain M_ntimes nK or Tin textEndV be diagonalizable. Then P_Ax decomposes as a product of linear i.e. deg factors P_Axlambda_-x...lambda_n-x where lambda_...lambda_n are the eigenvalues of A but there might be repetitions among the lambda_i's

Solution:
Proof. By astion exists an invertible matrix P s.t. P^-AP pmatrix lambda_ & & & ddots & & & lambda_n pmatrixtext denoted by Lambda. Longrightarrow P_AxP_Lambdaxtextdet pmatrix lambda_-x & & & ddots & & & lambda_n-x pmatrix lambda_-x...lambda_n-x
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Tags
eigenvalue, eigenvector, endomorphism, 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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