Unformatted text preview: be the diagonal entries of the normal form of xIA . For which matrices A is f 1 6 = 1? 12) Construct T with minimal polynomial x 2 ( x1) 2 and characteristic polynomial x 3 ( x1) 4 . Describe the primary decomposition of V under T and ﬁnd the projections on the primary components. 12) If N is a nilpotent linear operator on V , show that for any polynomial f the semisimple part of f ( N ) is a scalar multiple of I . 12) Let F be a subﬁeld of the complex numbers, V a ﬁnitedimensional vectorspace over F and T a semisimple operator on V . If f is any polynomial over F , prove that f ( T ) is semisimple. 12) Let T be a linear operator on ndimensional V over F a subﬁeld of C . Prove that T is semisimple iﬀ “if f is a polynomial over F and f ( T ) is nilpotent, then f ( T ) = 0.” 1...
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 Spring '09
 Linear Algebra, Matrices, Vector Space, Characteristic polynomial, Complex number

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