# B if the matrices a and b satisfy x ay x by for all

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b) If the matrices A and B satisfy ( X, AY ) = ( X, BY ) for all vectors X and Y , show that A = B . Solution: We have 0 = ( X, AY ) − ( X, BY ) = ( X, ( AY BY ) ) = ( X, ( A B ) Y ) for all X and Y so by part (a) with C := A B , we conclude that A = B . 12. [p. 9 #11–12] A matrix A is called anti-symmetric (or skew-symmetric) if A = A . a) Give an example of a 3 × 3 anti-symmetric matrix. Solution: The most general anti-symmetric 3 × 3 matrix has the form 0 a b a 0 c b c 0 . b) If A is any anti-symmetric matrix, show that ( X, AX ) = 0 for all vectors X . Solution: ( X, AX ) = ( A X, X ) = −( AX, X ) = −( X, AX ) . Thus 2 ( X, AX ) = 0 so ( X, AX ) = 0. 7

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c) Say X ( t ) is a solution of the differential equation dX dt = AX , where A is an anti- symmetric matrix. Show that bardbl X ( t ) bardbl = constant. [ Remark: A special case is that X ( t ) := parenleftbigg cos t sin t parenrightbigg satisfies X = AX with A = ( 0 1 1 0 ) so this problem gives another proof that cos 2 t + sin 2 t = 1]. Solution: Let E ( t ) := bardbl X ( t ) bardbl 2 . We show that dE/dt = 0. But, using part (b), dE dt = d dt ( X ( t ) , X ( t ) ) = 2 ( X ( t ) , X ( t ) ) = 2 ( X ( t ) , AX ( t ) ) = 0 . Bonus Problem [Please give this directly to Professor Kazdan] 1-B This is a followup to problem 7. a) If a projection P is self-adjoint, so P = P , show that P is an orthogonal projec- tion. b) Conversely, if P is an orthogonal projection, show that it is self-adjoint. [Last revised: November 10, 2012] 8
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