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# lec5-3 - Midterm Review Wednesday February 17 Open book No...

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Midterm Review Wednesday, February 17 Open book No notes Chapters 1-5, focus on lectures and homeworks Patrick K. Schelling Introduction to Theoretical Methods

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Eigenvalues and eigenvectors; diagonalization We have described linear operators acting on vectors in some space to yield new vectors Mr = k A special case occurs when k = λ r , where λ is just a number called an eigenvalue Mr = λ r We can write this in index notation as, X j M ij r j = λ r i The vector r in this case is the eigenvector corresponding to the eigenvalue λ For a square n × n matrix, there can be n eigenvalues and n corresponding eigenvectors Patrick K. Schelling Introduction to Theoretical Methods
Matrix diagonalization We can diagonalize a matrix M by making a matrix U with columns as the eigenvectors of A We can find U - 1 Then we have U - 1 MU = D , where D is a matrix of the eigenvalues Called a similarity transformation Patrick K. Schelling Introduction to Theoretical Methods

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Hermitian matrices Take A as the adjoint , and define it as A = ( A * ) T If A = A , the matrix is said to be Hermitian The eigenvalues of a Hermitian matrix are real, and the eigenvectors are orthogonal (and can be made orthonormal) Proof: Hr = λ r Take the adjoint of both sides to get equation r H = λ * r Patrick K. Schelling Introduction to Theoretical Methods
Proof, continued Multiply first equation by r on left, second equation by r on right, and combine them ( λ * - λ ) r r = 0 Now take two vectors r 1 , r 2 , with Hr 1 = λ 1 r 1 and Hr 2 = λ 2 r 2 We can multiply first equation on left by r 2 r 2 Hr 1 = λ 1 r 2 r 1 Multiply second equation on left by r 1 r 1 Hr 2 = λ 2 r 1 r 2 Patrick K. Schelling Introduction to Theoretical Methods

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Proof continued Take adjoint of second equation, and use H = H ( r 1 Hr 2 ) = r 2 Hr 1 = λ 1 r 2 r 1 = ( λ 2 r 1 r 2 ) = λ 2 r 2 r 1 So finally we have, using λ * 2 = λ 2 ( λ 1 - λ 2 ) r 2 r 1 = 0 So then r i r j = δ i , j for normalized vectors Patrick K. Schelling Introduction to Theoretical Methods
Similarity transformation (diagonalization) of Hermitian matrices When eigenvectors are orthonormal, then U is unitary U U = UU = I We can then say U - 1 = U , and U HU = D We can also left multiply by U , UU HU = HU = UD Patrick K. Schelling Introduction to Theoretical Methods

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Orthogonal transformation
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lec5-3 - Midterm Review Wednesday February 17 Open book No...

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