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lec3-2 - Unitary matrices We have already seen orthogonal...

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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
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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
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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
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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 0 4 Patrick K. Schelling Introduction to Theoretical Methods
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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 , 2-dimensions), 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
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Span, basis, and dimension example For example, take v 1 = 1 / 2 1 / 2 0 and v 2 = 1 / 2 - 1 / 2 0 The v 1 and v 2 are linearly independent and both lie in the x-y plane Any vector in the x-y plane can be written as a linear combination of v 1 and v 2 For example, V = 3 1 0 = a 1 v 1 + a 2 v 2 We notice that v 1 v 2 = ( 1 / 2 1 / 2 0 ) 1 / 2 - 1 / 2 0 , and also v 1 v 1 = v 2 v 2 = 1 (the v 1 and v 2 are orthonormal) Patrick K. Schelling Introduction to Theoretical Methods
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Example, continued Take 3 1 0 = a 1 v 1 + a 2 v 2 and left multiply by v 1 Then we can find a 1 a 1 = ( 1 / 2 1 / 2 0 ) 3 1 0 = 4
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