605 - Norm-preserving (or length-keeping) linear operator T...

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1 Norm-preserving (or length-keeping) linear operator T on R n : || T ( v ) || = || v || ∀ v R n . Example: T : linear operator on R 2 that rotates a vector by θ . T is norm-preserving. Example: U : linear operator on R n that has an eigenvalue λ≠ ± 1. U is not norm-preserving, since for the corresponding eigenvector v , || U ( v ) || = ||λ v || = |λ| · || v || ≠ || v || . Necessary conditions for a linear operator to be norm-preserving: Let Q = [ q 1 q 2 " q n ] be the standard matrix of the linear operator. Then (1) || q j || = || Q e j || = || e j || = 1, and (2) || q i + q j || 2 = || Q e i + Q e j || 2 = || Q ( e i + e j ) || 2 = || e i + e j || 2 = 2 = || q i || 2 + || q j || 2 , i.e., q i and q j are orthogonal. Example: is an orthogonal matrix. Proof (a) (b) with Q = [ q 1 q 2 " q n ], q i q i = 1 = [ Q T Q ] ii i , and q i q j = 0 = [ Q T Q ] ij i j . Q T Q = I n , and Q is invertible with Q 1 = Q T . (b) (c) u , v R n , Q u Q v = u Q T Q v = u Q 1 Q v = u v . ( Q T Q = I n .)
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2 (c) (d) u R n , || Q u || = ( Q u Q u ) 1/2 = ( u u ) 1/2 = || u || . (d) (a) The above necessary conditions. Corollary: (a) Q is orthogonal if and only if the rows Q of form an orthonormal basis of R n , or equivalently, QQ T = I n . (b) Q is orthogonal if and only if Q T is orthogonal. Proof Q T Q = I n QQ T = I n Q T = Q 1 ( Q T ) T Q T = I n . Proof (a) QQ T = I n 1 = det( I n ) = det( QQ T ) = det( Q )det( Q T ) = det( Q ) 2 . det( Q ) = ± 1. (b) ( PQ ) T = Q T P T = Q 1 P 1 = ( PQ ) 1 . (c) ( Q 1 ) T = ( Q T ) 1 = ( Q 1 ) 1 .
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This note was uploaded on 10/16/2010 for the course EE 155 taught by Professor Fong during the Spring '09 term at National Taiwan University.

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605 - Norm-preserving (or length-keeping) linear operator T...

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