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Chapter 13 Matrix Representation Chapter 13 Matrix Representation Matrix Rep. Same basics as introduced already. Convenient method of working with vectors. Superposition Complete set of vectors can be used to express any other vector. Complete set of N orthonormal vectors can form other complete sets of N orthonormal vectors. Can find set of vectors for Hermitian operator satisfying Au= α u. Eigenvectors and eigenvalues Matrix method Find superposition of basis states that are eigenstates of particular operator. Get eigenvalues. Copyright – Michael D. Fayer, 2018 Orthonormal basis set in N dimensional vector space { e j } basis vectors Any N dimensional vector can be written as N = j j x∑ xej with xj = ex j=1 To get this, project out eejj from x j jj j xej = e e x piece of x that is e , then sum over all e j . Copyright – Michael D. Fayer, 2018 Operator equation y= Ax NN jj ∑∑yejj= A xe Substituting the series in terms of bases vectors. jj=11= N j = ∑ xj Ae j=1 Left mult. by ei N ij yij= ∑ e Ae x j=1 The N2 scalar products ij e Ae N values of j for each yi; and N different yi are completely determined by Ae and the basis set { j }. Copyright – Michael D. Fayer, 2018 Writing = ij j aij e Ae Matrix elements of A in the basis { e } gives for the linear transformation N = yi∑ ax ij j jN= 12,, j=1 j Know the aij because we know A and { e } In terms of the vector representatives Q=++754 xyzˆˆˆ x1 y1 vector x2 y 7 x = y = 2 vector representative, 5 must know basis xN yN 4 (Set of numbers, gives you vector when basis is known.) The set of N linear algebraic equations can be written as y= Ax double underline means matrix Copyright – Michael D. Fayer, 2018 A array of coefficients - matrix aa11 12 a1N aa a 21 22 2 N The aij are the elements of the matrix A . Aa=( ij ) = aaNN12 aNN y= Ax vector representatives in particular basis y1 aa 11 12 a1N x 1 y aa x 2 = 21 22 2 yNNN aa12 aNN x N The product of matrix A and vector representative x is a new vector representative y with components N yi= ∑ ax ij j j=1 Copyright – Michael D. Fayer, 2018 Matrix Properties, Definitions, and Rules Two matrices, A and B are equal AB= if aij = bij. The unit matrix The zero matrix 10 0 01 0 ones down 00 0 1 =δ = ij principal diagonal 00 0 0 = 00 1 00 0 Gives identity transformation N 00x = yi =∑δ ij xx j = i j=1 Corresponds to y=1 xx = Copyright – Michael D. Fayer, 2018 Matrix multiplication Consider y= Ax z= By operator equations z= BA x Using the same basis for both transformations N = zk∑ by ki i z= By Bb= matrix = (ki ) i=1 z= By = BAx = Cx C= BA has elements Example N c= ba 23 75 2928 kj∑ ki ij = i=1 34 56 4139 Law of matrix multiplication Copyright – Michael D. Fayer, 2018 Multiplication Associative ( AB) C= A( BC) Multiplication NOT Commutative except in special cases. AB ≠ BA Matrix addition and multiplication by complex number αβA+= BC cabij=αβ ij + ij Copyright – Michael D. Fayer, 2018 Reciprocal of Product −1 −− ( AB) = B11 A For matrix defined as Aa= ()ij Transpose Aa= ( ji ) interchange rows and columns Complex Conjugate * * Aa= ( ij ) complex conjugate of each element Hermitian Conjugate + * Aa= ( ji ) complex conjugate transpose Inverse of a matrix A − −−11 inverse of A A 1 AA= A A = 1 identity matrix CT − A transpose of cofactor matrix (matrix of signed minors) A 1 = A determinant A≠=0 If AA 0 is singular Copyright – Michael D. Fayer, 2018 Rules ( AB) = BA transpose of product is product of transposes in reverse order ||||AA = determinant of transpose is determinant ** ()AB* = A B complex conjugate of product is product of complex conjugates |AA* |||= * determinant of complex conjugate is complex conjugate of determinant ++ ()AB+ = B A Hermitian conjugate of product is product of Hermitian conjugates in reverse order + |AA |||= * determinant of Hermitian conjugate is complex conjugate of determinant Copyright – Michael D. Fayer, 2018 Definitions AA= Symmetric + AA= Hermitian AA= * Real * AA= − Imaginary −+1 AA= Unitary aaij= ijδ ij Diagonal Powers of a matrix A01=1 A = A A 2 = AA ⋅⋅⋅ A2 eAA =1 + + +⋅⋅⋅ 2! Copyright – Michael D. Fayer, 2018 Column vector representative one column matrix x1 x x = 2 vector representatives in particular basis xN then y= Ax becomes y1 aa 11 12 a1N x 1 y aa x 2 = 21 22 2 yNNN aa12 aNN x N row vector transpose of column vector x = ( xx12, xN ) y= Ax y= xA transpose + y= Ax y++= xA Hermitian conjugate Copyright – Michael D. Fayer, 2018 Change of Basis orthonormal basis { ei } then ij ee= δ ij ( ij,= 1, 2, N) Superposition of { ei } can form N new vectors linearly independent a new basis { ei′ } N ik′ e= ∑ ueik i= 1, 2, N k=1 complex numbers Copyright – Michael D. Fayer, 2018 New Basis is Orthonormal ji′′ ee = δ ij if the matrix Uu= ( ik ) coefficients in superposition N ik′ e= ∑ ueik i= 1, 2, N meets the condition k=1 + UU= 1 −+ UU1 = U is unitary – Hermitian conjugate = inverse Important result. The new basis { e i′ } will be orthonormal if U , the transformation matrix, is unitary (see book ++ and Errata and Addenda, linear algebra book ). UU= U U = 1 Copyright – Michael D. Fayer, 2018 Unitary transformation substitutes orthonormal basis { e ′ } for orthonormal basis { e } . Vector x i x= ∑ xei i vector – line in space (may be high dimensionality abstract space) ′ i′ x= ∑ xei written in terms of two basis sets i x Same vector – different basis. The unitary transformation U can be used to change a vector representative of x in one orthonormal basis set to its vector representative in another orthonormal basis set. x – vector rep. in unprimed basis x' – vector rep. in primed basis x′ = Ux change from unprimed to primed basis + x= Ux′ change from primed to unprimed basis Copyright – Michael D. Fayer, 2018 Example Consider basis {xyzˆˆ,,ˆ} y |s z x Vector s - line in real space. In terms of basis s=++771 xˆˆ yzˆ Vector representative in basis {xyzˆˆ,,ˆ} 7 = s 7 1 Copyright – Michael D. Fayer, 2018 Change basis by rotating axis system 45° around zˆ . Can find the new representative of s , s' s′ = Us U is rotation matrix x yz cosθ sinθ 0 U = −sinθ cosθ 0 0 01 For 45° rotation around z 2/2 2/2 0 U = − 2/2 2/2 0 0 01 Copyright – Michael D. Fayer, 2018 Then 2/2 2/2 0 7 72 ′ s =−=2/2 2/2 0 7 0 0 0 11 1 72 s e′ s′ = 0 vector representative of in basis { } 1 Same vector but in new basis. Properties unchanged. 1/2 Example – length of vector ( ss) 1/2 1/2 1/2 1/2 ss=⋅( s* s ) = (49 ++ 49 1) = (99) 1/2 * 1/2 1/2 1/2 ss=( s′′ ⋅ s ) = (2 × 49 ++ 0 1) = (99) Copyright – Michael D. Fayer, 2018 Can go back and forth between representatives of a vector x by change from unprimed change from primed to primed basis to unprimed basis + x′ = Ux x= Ux′ components of x in different basis Copyright – Michael D. Fayer, 2018 Consider the linear transformation y= Ax operator equation In the basis { e } can write y= Ax or yi= ∑ ax ij j j Change to new orthonormal basis { e ′ } using U + y′′= Uy = UAx = UAUx or y′′= Ax′ with the matrix A ′ given by + A′ = U AU −1 Because U is unitary A′ = U AU Copyright – Michael D. Fayer, 2018 Extremely Important − A′ = U AU 1 Can change the matrix representing an operator in one orthonormal basis into the equivalent matrix in a different orthonormal basis. Called Similarity Transformation Copyright – Michael D. Fayer, 2018 yAx= ABC = ABC += In basis { e } Go into basis { e′ } yAx′′=′ ABC′′= ′ ABC ′ += ′ ′ Relations unchanged by change of basis. Example AB= C ++ U ABU= UCU + +− Can insert UU between AB because UU = U1 U = 1 ++ + U AU U BU= U CU Therefore AB′′= C ′ Copyright – Michael D. Fayer, 2018 Isomorphism between operators in abstract vector space and matrix representatives. Because of isomorphism not necessary to distinguish abstract vectors and operators from their matrix representatives. The matrices (for operators) and the representatives (for vectors) can be used in place of the real things. Copyright – Michael D. Fayer, 2018 Hermitian Operators and Matrices Hermitian operator xAy= yAx Hermitian operator Hermitian Matrix + AA= + - complex conjugate transpose - Hermitian conjugate Copyright – Michael D. Fayer, 2018 Theorem (Proof: Powell and Craseman, P. 303 – 307, or linear algebra book) For a Hermitian operator A in a linear vector space of N dimensions, there exists an orthonormal basis, UU12, UN relative to which A is represented by a diagonal matrix α1 00 00α A′ = 2 0 . 0 α N i The vectors, U , and the corresponding real numbers, αi, are the solutions of the Eigenvalue Equation AU= α U and there are no others. Copyright – Michael D. Fayer, 2018 Application of Theorem Operator A represented by matrix A in some basis { e i } . The basis is any convenient basis. In general, the matrix will not be diagonal. There exists some new basis eigenvectors { U i } in which A ′ represents operator and is diagonal eigenvalues. To get from { e i } to { U i } unitary transformation. { Uii} = Ue{ }. − A′ = U AU 1 Similarity transformation takes matrix in arbitrary basis into diagonal matrix with eigenvalues on the diagonal.
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