Ans 4 - v1

# Ans 4 - v1 - Copyright 2005 by the Society for Industrial...

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“ajlbook” — 2004/11/9 — 13:36 — page 139 — #147 Chapter 13 Kronecker Products 13.1 Deﬁnition and Examples Deﬁnition 13.1. Let A R m × n , B R p × q . Then the Kronecker product (or tensor product) of A and B is deﬁned as the matrix A B = a 11 B ··· a 1 n B . . . . . . . . . a m 1 B a mn B R mp × nq . (13.1) Obviously, the same deﬁnition holds if A and B are complex-valued matrices. We restrict our attention in this chapter primarily to real-valued matrices, pointing out the extension to the complex case only where it is not obvious. Example 13.2. 1. Let A = ± 123 321 ² and B = ± 21 23 ² . Then A B = ³ B 2 B 3 B 3 B 2 BB ´ = 214263 234669 634221 694623 . Note that B A ±= A B . 2. For any B R p × q , I 2 B = ± B 0 0 B ² . Replacing I 2 by I n yields a block diagonal matrix with n copies of B along the diagonal. 3. Let B be an arbitrary 2 × 2 matrix. Then B I 2 = b 11 0 b 12 0 0 b 11 0 b 12 b 21 0 b 22 0 0 b 21 0 b 22 . 139 Copyright ©2005 by the Society for Industrial and Applied Mathematics This electronic version is for personal use and may not be duplicated or distributed. From "Matrix Analysis for Scientists and Engineers" Alan J. Laub. Buy this book from SIAM at www.ec-securehost.com/SIAM/ot91.html

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“ajlbook” — 2004/11/9 — 13:36 — page 140 — #148 140 Chapter 13. Kronecker Products The extension to arbitrary B and I n is obvious. 4. Let x R m , y R n . Then x y = ± x 1 y T ,...,x m y T ² T = [ x 1 y 1 1 y n ,x 2 y 1 m y n ] T R mn . 5. Let x R m , y R n . Then x y T = [ x 1 y,. ..,x m y ] T = x 1 y 1 ... x 1 y n . . . . . . . . . x m y 1 x m y n = xy T R m × n . 13.2 Properties of the Kronecker Product Theorem 13.3. Let A R m × n , B R r × s , C R n × p , and D R s × t . Then (A B)(C D) = AC BD ( R mr × pt ). (13.2) Proof: Simply verify that (A B)(C = a 11 B ··· a 1 n B . . . . . . . . . a m 1 B a mn B c 11 D c 1 p D . . . . . . . . . c n 1 D c np D = n k = 1 a 1 k c k 1 BD n k = 1 a 1 k c kp . . . . . . . . . n k = 1 a mk c k 1 n k = 1 a mk c = AC BD. Theorem 13.4. For all A and B , (A B) T = A T B T . Proof: For the proof, simply verify using the deﬁnitions of transpose and Kronecker product. Corollary 13.5. If A R n × n and B R m × m are symmetric, then A B is symmetric. Theorem 13.6. If A and B are nonsingular, (A 1 = A 1 B 1 . Proof: Using Theorem 13.3, simply note that (A B)(A 1 B 1 ) = I I = I. Copyright ©2005 by the Society for Industrial and Applied Mathematics This electronic version is for personal use and may not be duplicated or distributed. From "Matrix Analysis for Scientists and Engineers" Alan J. Laub. Buy this book from SIAM at www.ec-securehost.com/SIAM/ot91.html
“ajlbook” — 2004/11/9 — 13:36 — page 141 — #149 13.2. Properties of the Kronecker Product 141 Theorem 13.7. If A R n × n and B R m × m are normal, then A B is normal.

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Ans 4 - v1 - Copyright 2005 by the Society for Industrial...

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