Applied linear algebra by Riaz A Usmani

By Riaz A Usmani

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Rn Á Rn ˆ 1; and Ri Á Rj ˆ 0; for i Tˆ j Thus the rows of A are unit vectors and are orthogonal to each other; hence they form an orthonormal set of vectors. The condition AT A ˆ I similarly shows that the columns of A also form an orthonormal set of vectors. Furthermore, since each step is reversible, the converse is true. This proves the theorem. 33. Consider the following block matrices (which are partitions of the same matrix): 2 (a) 1 À2 0 1 42 3 5 7 3 1 4 5 3 3 À2 5, 9 2 (b) 1 À2 0 42 3 5 3 1 4 3 1 3 7 À2 5 5 9 Find the size of each block matrix and also the size of each block.

The inner numbers 3 and 2 are not equal ; hence the product BA is not de®ned. Lipschutz−Lipson:Schaum’s Outline of Theory and Problems of Linear Algebra, 3/e 2. Algebra of Matrices CHAP. 7. © The McGraw−Hill Companies, 2004 Text 2 Find AB, where A ˆ 4 3 À2  À1 5 2 2 À1 and B ˆ 4 1 3 4 1 3 6 1 5. 2 0 À5 À2 Since A is a 2  3 and B a 3  4 matrix, the product AB is de®ned and is a 2  4 matrix. 8. Find: (a) (a) 1 À3 6 5   2 , À7  (b) 2 À7   6 , 5 1 À3  (c) ‰2; À7Š  1 6 . À3 5 The ®rst factor is 2  2 and the second is 2  1, so the product is de®ned as a 2  1 matrix:        1 6 2 2 À 42 À40 ˆ ˆ À3 5 À7 À6 À 35 À41 (b) The product is not de®ned, since the ®rst factor is 2  1 and the second factor is 2  2.

Find a 3  3 orthogonal matrix P whose ®rst two rows are multiples of: (a) …1; 2; 3† and …0; À2; 3†, (b) …1; 3; 1† and …1; 0; À1†. 73. Suppose A and B are orthogonal matrices. Show that AT , AÀ1 , AB are also orthogonal. 74.   3 À4 1 Which of the following matrices are normal? A ˆ ,Bˆ 4 3 2 COMPLEX MATRICES 2  1 À2 , C ˆ 40 3 0 2 3 1 1 1 1 5. 75. 76. Suppose A is a complex matrix. Show that AAH and AH A are Hermitian. 77. Let A be a square matrix. Show that: (a) A ‡ AH is Hermitian, (b) (c) A ˆ B ‡ C, where B is Hermitian and C is skew-Hermitian.

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