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Tag Archives: orthogonal matrices
Linear Algebra and Its Applications, Exercise 3.4.13
Exercise 3.4.13. Given the vectors and the matrix whose columns are , , and , use GramSchmidt orthogonalization to factor . Answer: We first choose . We then have We then have We have , so , , and . … Continue reading
Posted in linear algebra
Tagged orthogonal matrices, orthogonalization, orthonormal vectors
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Linear Algebra and Its Applications, Exercise 3.4.12
Exercise 3.4.12. Given the vectors and , find a scalar such that is orthogonal to . Given the matrix whose columns are and respectively, find matrices and such that is orthogonal and . Answer: We must have . This implies … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.11
Exercise 3.4.11. If the matrix is both upper triangular and orthogonal, show that must be a diagonal matrix. Answer: Let be an by matrix. Since is upper triangular we have where for . Our goal is to prove that is … Continue reading
Posted in linear algebra
Tagged diagonal matrix, orthogonal matrices, upper triangular matrix
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Linear Algebra and Its Applications, Exercise 3.4.6
Exercise 3.4.6. Given the matrix find entries for the third column such that is orthogonal. How much freedom do you have to choose the entries? Finally, verify that both the columns and rows are orthonormal. Answer: In order for to … Continue reading
Linear Algebra and Its Applications, Exercise 3.4.4
Exercise 3.4.4. Given two orthogonal matrices and , show that their product is also orthogonal. If represents rotation through the angle and represents rotation through the angle , what does represent? What trigonometric identities for and can be found in … Continue reading