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554-HW-12q

# 554-HW-12q - be the diagonal entries of the normal form of...

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12) Classify up to similarity all matrices A C 3 × 3 such that A 3 = I . 12) For n N 2 let N be an n × n matrix over a field F such that N n = 0 but N n - 1 6 = 0. Show that N has no square root, i.e. there is no A F n × n such that A 2 = N . 12) If N is a nilpotent matrix in C 3 × 3 then prove that A = I + 1 2 N - 1 8 N 2 satisfies A 2 = I + N . Use the binomial expansion formula on (1 + x ) 1 / 2 to obtain a similar formula for a square root of I + N for N a nilpotent matrix over C n × n . 12) Use 15 to prove that for nonzero c C and nilpotent N C n × n we know ( cI + N ) has a square root. Then use Jordan form to prove that non-singular matrices C n × n have square roots. 12) True or false: every matrix in F [ x ] n × n is row-equivalent to an upper-triangular matrix? 12) Let T be a linear operator on the n -dimensional vectorspace V . Let A be the matrix representation of T in the ordered basis B . Show that T has a cyclic vector iff the determinants of the ( n - 1) × ( n - 1) submatrices of xI - A are relatively prime. 12) Let A be an n × n matrix with entries in the field F and let f 1 , f 2 , · · ·
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Unformatted text preview: be the diagonal entries of the normal form of xI-A . For which matrices A is f 1 6 = 1? 12) Construct T with minimal polynomial x 2 ( x-1) 2 and characteristic polynomial x 3 ( x-1) 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 semi-simple part of f ( N ) is a scalar multiple of I . 12) Let F be a subﬁeld of the complex numbers, V a ﬁnite-dimensional vectorspace over F and T a semi-simple operator on V . If f is any polynomial over F , prove that f ( T ) is semi-simple. 12) Let T be a linear operator on n-dimensional V over F a subﬁeld of C . Prove that T is semi-simple iﬀ “if f is a polynomial over F and f ( T ) is nilpotent, then f ( T ) = 0.” 1...
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