Solution: Exercises 15 and 16 concern the (real) Schur

Chapter , Problem 16E

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Exercises 15 and 16 concern the (real) Schur factorization of an n × n matrix A in the form A = URUT, where U is an orthogonal matrix and R is an n × n upper triangular matrix.Let A be an n × n matrix with n real eigenvalues, counting multiplicities, denoted by . It can be shown that A admits a (real) Schur factorization. Parts (a) and (b) show the key ideas in the proof. The rest of the proof amounts to repeating (a) and (b) for successively smaller matrices, and then piecing together the results.a. Let u1 be a unit eigenvector corresponding to , let be any other vectors such that is an orthonormal basis for Rn, and then let U = . Show that the first column of is the first column of the n × n identity matrix.b. Part (a) implies that UTAU has the form shown below. Explain why the eigenvalues of . [Hint: See the Supplementary Exercises for Chapter 5.]