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**Unformatted text preview: **A = ± r cos θ-r sin θ r sin θ r cos θ ² ∈ R 2 × 2 . Here r > 0. Problem #5. Let A = ± 13 9-16 37 ² and B = ±-35 45 20 10 ² . a) Compute the Schur factorization of A and of B . (Hint: Use the proof of Theorem 24.9 of Trefethen-Bau and the fact that the characteristic polynomials of A and B are easy to factor to ﬁnd these factorizations). b) Determine, for both A and B , whether the matrix is diagonalizable. If it is not explain why not and if it is diagonalize it. Bonus Problem. We say a matrix A ∈ C m × m is normal if A * A = AA * . Show using the Schur factorization that if A is normal then A is unitarily diagonalizable. That is A = Q Λ Q * for Λ diagonal and Q unitary....

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