Explain why the singular values of A are the same as the nonzero eigenvalues of the positive definite square root matrix S = √ATA , defined in Exercise 8.5.27.

Data From Exercise 8.5.27

(a) Prove that every positive definite matrix K has a unique positive definite square  root, i.e., a matrix B > 0 satisfying B² = K.(b) Find the positive definite square roots of the following matrices:

(i)

(ii)

(iii)

(iv)

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