Practice Algebra Qualifying Exam Solutions
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چکیده
1. Let A be an n× n matrix with complex coefficients. Define trA to be the sum of the diagonal elements. Show that trA is invariant under conjugation, i.e., trA = trPAP for all invertible n× n matrices P. Proof. Let P be an invertible matrix. Let ~pk be the k-th row of P, ~qj the j-th column of P, and ~ ai the i-th column of A. The k-th row of the matrix PA is 〈~pk · ~ a1, . . . ,~pk · ~ an〉. So the k,k-th entry of PAP is 〈~pk · ~ a1, . . . ,~pk · ~ an〉 · ~qk. Hence
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