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

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Unformatted text preview: Midterm Review Wednesday, February 17 Open book No notes Chapters 1-5, focus on lectures and homeworks Patrick K. Schelling Introduction to Theoretical Methods 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 = r , where is just a number called an eigenvalue Mr = r We can write this in index notation as, X j M ij r j = r i The vector r in this case is the eigenvector corresponding to the eigenvalue 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 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 = r Take the adjoint of both sides to get equation r H = * 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 ( *- ) r r = 0 Now take two vectors r 1 , r 2 , with Hr 1 = 1 r 1 and Hr 2 = 2 r 2 We can multiply first equation on left by r 2 r 2 Hr 1 = 1 r 2 r 1 Multiply second equation on left by r 1 r 1 Hr 2 = 2 r 1 r 2 Patrick K. Schelling Introduction to Theoretical Methods Proof continued Take adjoint of second equation, and use H = H ( r 1 Hr 2 ) = r 2 Hr 1 = 1 r 2 r 1 = ( 2 r 1 r 2 ) = 2 r 2 r 1 So finally we have, using * 2 = 2 ( 1- 2 ) r 2 r 1 = 0 So then r i r j = 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 Orthogonal transformation...
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This note was uploaded on 07/30/2011 for the course PHZ 3113 taught by Professor Staff during the Spring '03 term at University of Central Florida.

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

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