Show that each of these is diagonalizable. (a) (b) x,y,z scalars 1 1
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(a)
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(b)
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x,y,z scalars
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a Using the formula for the inverse of a 2 2 matrix gives this Now pick scalars ad so that ad bc 0 and 2d 2 2c 2 0 and 2a 2 2b 2 0 For example these will do b As above we are looking for scalars ad so that ad bc 0 and abx b 2 y a 2 y abz 0 and cdx d 2 y c 2 y cdz 0 no matter what values x y and z have For starters we assume that y 0 else the given matrix is already diagonal We shall use that assumption because if we arbitrarily let a 1 then we get bx b 2 y y bz 0 yb 2 z xb y 0 and the quadratic formula gives note that if x y and z are real then these two bs are real as the discriminant is positive By the same token if we arbitrarily let c 1 then dx d 2 y y dz 0 yd 2 x zd y 0 and we get here as above if x y z 2 R then this discriminant is positive so a symmetric real 2 2 matrix is similar to a real diagonal matrix For a check we try x 1 y 2 z 1 Not all four choices bd 11 11 satisfy ad bc 0 a ...View the full answer
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