Math 6610 • Analysis of Numerical Methods I

**Matrix Factorizations** **a)** Prove any two of the following statements: **(i) Schur Decomposition:** Any matrix \(A\in\mathbb{C}^{m\times m}\) can be factored as \(A=Q^*TQ\), where \(Q\) is unitary and \(T\) is upper triangular. **(ii) Singular Value Decomposition:** Any matrix \(A\in\mathbb{C}^{m\times n}\) can be factored as \(A=U\Sigma V^*\), where \(U\in\mathbb{C}^{m\times m}\) and \(V\in\mathbb{C}^{n\times n}\) are unitary and \(\Sigma\in\mathbb{R}^{m\times n}\) is a rectangular matrix whose only nonzero entries are non-negative entries on its diagonal. **(iii) QR Factorization:** Any full-rank matrix \(A\in\mathbb{R}^{m\times n}\) for \(m\geq n\) can be factored \(A=QR\) where \(Q\in\mathbb{R}^{m\times m}\) is orthogonal and \(R\in\mathbb{R}^{m\times n}\) is upper triangular with positive diagonal entries. **b)** Discuss situations in which each of these factorizations is useful in numerical analysis and explain why the factorizations are useful in those situations.