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HW1_602_2011

# HW1_602_2011 - primitive n th root of unity(iii Why do we...

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Math 602 HOMEWORK 1 1. (a) Show that the product of two unitary matrices is also a unitary matrix. (b) If U is unitary, is its inverse U - 1 unitary? (c) Is the identity matrix unitary? 2. (a) Show that the determinant of a unitary matrix has absolute value 1. (b) What is the determinant of an orthogonal matrix? 3. Let T be a complex matrix. (a) Write the eigenvalues of T * in terms of the eigenvalues of T . (b) Write the determinant of T * in terms of the determinant of T . (c) Write the trace of T * in terms of the trace of T . 4. (a) Find all the 2 × 2 unitary matrices. (b) Find all the 2 × 2 orthogonal matrices. 5. a) Recall the roots of 1: (i) Show that the n complex solutions of the equation z n = 1 are z k = exp[2 πik/n ] , k = 0 , 1 , . . . , n - 1 (called the n th roots of unity ). (ii) Show that z k = z k 1 . ( z 1 is called the
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Unformatted text preview: primitive n th root of unity .) (iii) Why do we stop the index at k = n-1? What is z n , z n +1 ? (iv) Plot these n solutions z , z 1 ,...,z n-1 in the the complex plane for (i) n = 2; (ii) n = 3; (iii) n = 4; (iv) for a general n . (Use the one separate plane for each n ). b) Let w = e 2 πi/n be the primitive n th root of unity. Consider the matrix U = 1 √ n [ w jk ] j,k =0 , 1 ,...,n-1 which is, in fact, the Fourier Matrix: U = 1 √ n 1 1 1 ··· 1 1 w w 2 ··· w n-1 1 w 2 w 4 ··· w 2( n-1) . . . . . . . . . . . . 1 w n-1 w 2( n-1) ··· w ( n-1) 2 TRUE or FALSE: U is unitary. Justify your answer. 1...
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