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Unformatted text preview: Unitary matrices We have already seen orthogonal matrices ( A T = A 1 where A is real), Hermitian matrices ( A = A ), and symmetric matrices ( A T = A where A is real) Another important kind of matrix is unitary defined by U = U 1 An important example is the matrix of eigenvectors U we obtain from solving the eigenvalue problem When we diagonlize a Hermitiam matrix H , we wrote this as U 1 HU = D However, it is easy to show (since the eigenvectors of H are orhonormal) that U 1 = U (it is unitary) Hence we find U U = UU = I , and U HU = D Patrick K. Schelling Introduction to Theoretical Methods Example, diagaonlizing a Hermitian matrix with a unitary similarity transformation Consider the example 2 in the book in Section 11, diagaonlizing a Hermitian matrix H , H = 2 3 i 3 + i 1 If we take H = ( H * ) T ( H * ) T = 2 3 + i 3 i 1 T = 2 3 i 3 + i 1 Hence we see that H is Hermitian ( H = H ) Eigenvalues found from (2 )( 1 ) 10 = 2  12 = 0 Find 1 = 3 and 2 = 4 Patrick K. Schelling Introduction to Theoretical Methods Example continued We find the eigenvectors for 1 = 3 by solving 5 3 i 3 + i 2 v 1 v 2 1 = 0 We find the eigenvector v 1 v 2 1 = 2 / 14 (3 + i ) / 14 For 2 = 4, we find v 1 v 2 2 = (3 i ) / 14 2 / 14 So we make the U matrix from the eigenvectors U = 2 / 14 (3 i ) / 14 ( 3 i ) / 14 2 / 14 Patrick K. Schelling Introduction to Theoretical Methods Adjoint operator U Given the U , we can make U U = 2 / 14 (3 i ) / 14 ( 3 i ) / 14 2 / 14 U = 2 / 14 ( 3 + i ) / 14 (3 + i ) / 14 2 / 14 Then we can easily show U U = I In this case, this happens because we diagonalized a Hermitian matrix Finally, you can easily verify U HU = D = 3 0 4 Patrick K. Schelling Introduction to Theoretical Methods Span, basis, and dimension If we have n linearly independent vectors of length n , v m ( m = 1 , 2 , 3 ,... n ), we can make any vector V of length n as a linear combination of the v m In this case, the set of vectors v m is said to span V n , with n the dimensionality of the space For example, two vectors v 1 and v 2 (that are not parallel, and hence are linearly independent) define a plane ( V 2 , 2dimensions), and any vector V lying in this plane can be written as a linear combination of v 1 and v 2 Patrick K. Schelling Introduction to Theoretical Methods Span, basis, and dimension example For example, take v 1 = 1 / 2 1 / 2 and v 2 = 1 / 2 1 / 2 The v 1 and v 2 are linearly independent and both lie in the xy plane Any vector in the xy plane can be written as a linear combination of v 1 and v 2 For example, V = 3 1 = a 1 v 1 + a 2 v 2 We notice that v 1 v 2 = ( 1 / 2 1 / 2 0 ) 1 / 2 1 / 2...
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 Spring '03
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