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18.06 Problem Set 7 - Solutions
Problem 5: (10=5+5) (a) If A (an n × n matrix) has n nonnegative eigenvalues λ k and independent eigenvectors x k, and if we define “ √ A” as the matrix with eigenvalues √ λ k and the same eigenvectors, show that (√ A)2 = A. Solution We can decompose A into A = SΛS−1, where S is the matrix consists of eigenvectors of A, and Λ = λ
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