22_hwc1SolnsODDA

22_hwc1SolnsODDA - J 2 is the matrix representing a...

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(b) For any numbers x and y , form the matrix xI + yJ . Show that the product of any two such matrices is another matrix of the same form. More specifically, ( xI + yJ )( uI + vJ ) = ( xu - yv ) I + ( xv + yu ) J . (c) Find all values of x and y so that ( xI + yJ ) 2 = 4 I . (d) Find all values of x and y so that ( xI + yJ ) 2 = - 4 I . The point of this problem is that if we identify the real numbers with the real multiples of I ; i.e., the matrices of the form xI , then in the class of matrices of the form uI + vJ there is a square root of x even if x is negative. We can identify the set of matrices of the from xI + yJ with the complex numbers, and what we have just given amounts to a construction of the complex number system out of the real numbers, using matrices. SOLUTION (a) Computing by the rules, J e 1 = e 2 and J e 2 = - e 1 . Notice that e 2 is what you get when you rotate e 1 counterclockwise through an angle of π/ 2, and that - e 1 is what you get when you rotate e 2 counterclockwise through an angle of π/ 2. Therefore, J is the matrix representing this rotation. It now follows that
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Unformatted text preview: J 2 is the matrix representing a counterclockwise rotation through the angle . But such a rotation sends any vector x into its opposite-x , so that J 2 =-I as you can see by direct calculation. For (b) note that xI + yJ = x-y y x and uI + vJ = u-v v u . By direct calculation, we nd x-y y x u-v v u = xu-yv-xv-yu xv + yu xu-yv = ( xu-yv ) I + ( xv + yu ) J . For (c) , if ( xI + yJ ) 2 = 4 I , then x 2-y 2-2 xy 2 xy x 2-y 2 = 4 4 . Looking at the o diagonal entries, we see that either x = 0 or y = 0. But if x = 0, then the diagonal entries would be-y 2 , which cannot equal 4. Hence it must be that y = 0 and x 2 = 4, or x = 2. Part (d) is almost the same: If ( xI + yJ ) 2 =-4 I , then x 2-y 2-2 xy 2 xy x 2-y 2 = -4-4 . 21 /september/ 2005; 22:09 23...
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This note was uploaded on 10/03/2010 for the course MATH 380 taught by Professor Gladue during the Fall '08 term at Roger Williams.

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