This preview shows pages 1–8. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Midterm Review Wednesday, February 17 Open book No notes Chapters 15, 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...
View
Full
Document
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.
 Spring '03
 Staff

Click to edit the document details