Q1. -Diagonalizability Let A E Mn (C).Recall hat A is said to be diagonalizable
ID: 3117437 • Letter: Q
Question
Q1. -Diagonalizability Let A E Mn (C).Recall hat A is said to be diagonalizable if there exists and invernible matrisx P such that PAP-i a diagonal matrixs. Determine which of the following statements are true (a) If A has a Jordan basis consisting only of chains of length 1, then A is diagonalizable. (b) If an invertible matrix P such that f -P-1 DP, P-1 DP , where D is diagonal, then A is diagonal. where is diagonal, then A is diagonalizable. (c) Ifan invertible matrix P such that A- (d) If A is diagonalizable, then a Jordan normal form of A must only have blocks of size 1Explanation / Answer
Option (c) is correct.
An n × n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. In fact, A = P-1DP , with D = PAP-1 a diagonal matrix, if and only if the columns of P-1 are n linearly independent eigenvectors of A. In this case, the diagonal entries of D are eigenvalues of A that correspond, respectively, to the eigenvectors in P.
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