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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 λ )( 1 λ ) 10 = λ 2 λ 12 = 0 • Find λ 1 = 3 and λ 2 = 4 Patrick K. Schelling Introduction to Theoretical Methods Example continued • We find the eigenvectors for λ 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 λ 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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