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Monthly Archives: May 2011
Linear Algebra and Its Applications, Exercise 1.6.18
Exercise 1.6.18. Suppose that Show that Answer: We have The product of two upper triangular matrices is also an upper triangular matrix, and multiplying by a diagonal matrix preserves this. The left side of the final equation above is therefore … Continue reading
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Linear Alegbra and Its Applications, Exercise 1.6.17
Exercise 1.6.17. (a) Suppose that the n by n matrix A can be factored as LDU, where L and U have ones on the diagonal. Factor the transpose of A. (b) If we have what triangular systems will provide a … Continue reading
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Linear Algebra and Its Applications, Exercise 1.6.16
Exercise 1.6.16. (i) If A is an n by n symmetric matrix, how many entries of A can be chosen independently of each other? (ii) If If K is an n by n skewsymmetric matrix, how many entries of K … Continue reading
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Linear Algebra and Its Applications, Exercise 1.16.15
Exercise 1.6.15. For any square matrix B and matrices A and K where prove that A is symmetric and K is skewsymmetric, i.e., For the case where compute A and K and show that B can be expressed as the … Continue reading
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Linear Algebra and Its Applications, Exercise 1.6.14
Exercise 1.6.14. For any m x n matrix A, prove that and are symmetric matrices. Provide an example where these matrices are not equal. Answer: Per Equation 1M(i) on page 47 we have Substituting for B we have Since is … Continue reading
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Linear Algebra and Its Applications, Exercise 1.16.13
Exercise 1.6.13. Compute , , , and for the following matrices: Answer: We have Note that per Equation 1M(i) on page 47 we have This means that and as confirmed by the computation above. NOTE: This continues a series of … Continue reading
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Linear Algebra and Its Applications, Exercise 1.6.12
Exercise 1.6.12. Suppose that A is an invertible matrix and has one of the following properties: (1) A is a triangular matrix. (2) A is a symmetric matrix. (3) A is a tridiagonal matrix. (4) All the entries of A … Continue reading
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