1Matrix Transformations Using Matrix Transformations Using EigenvectorsEigenvectorsLarry CarettoMechanical Engineering 501ASeminar in Engineering Seminar in Engineering AnalysisAnalysisSeptember 9, 20042Outline• Review last lecture• Transformations with a matrix of eigenvectors: Λ = X-1AX• Hermitian and orthogonal matrices• Quadratic forms• Numerical methods for eigenvalues and eigenvectors3Review Eigens• Basic definition (A n x n):Ax= λx•Det[A – Iλ] = 0 gives nthorder equation for eigenvalues– n eigenvalues (may not be distinct)–solve [A – Iλ]x = 0 for n components of each of n eigenvectors – eigenvectors undetermined to within a multiplicative constant– eigenvectors may or may not be linearly independent4Transform A into a Diagonal• If the n eigenvectors of A are linearly independent we can define an invertible matrix, X, whose columns are eigen-vectors of A: X = [x(1) x(2) x(3) ….. x(n)]• AX = [Ax(1) Ax(2) Ax(3) ….. Ax(n)]• AX = [λ1x(1) λ2x(2) λ3x(3) ….. λnx(n)]• We now show that AX = ΛD where Λ is a diagonal matrix of eigenvalues5Matrix Product XΛ⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡=nnnnnnnnnxxxxxxxxxxxxxxxxλλλλLLMOMMMMOMMMLLLLLLLLMOMMMMOMMMLLLLLL000000000000321)()3()2()1(3)(3)3(3)2(3)1(2)(2)3(2)2(2)1(1)(1)3(1)2(1)1(XΛ• Usual X matrix component is, xrow,column,•This X component notation is x(vector)row• Usual matrix multiplication formula applies6Matrix Product XΛ Continued[]AXxxxxXΛ==⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡=)(1)3(1)2(1)1(1)()3(3)2(2)1(13)(3)3(33)2(23)1(12)(2)3(32)2(22)1(11)(1)3(31)2(21)1(1nnnnnnnnnnnnnxxxxxxxxxxxxxxxxλλλλλλλλλλλλλλλλλλλλLLLLMOMMMMOMMMLLLLLL• AX = [λ1x(1) λ2x(2) λ3x(3) ….. λnx(n)] from previous slide. We now see that AX = XΛ27A Transformed• We assumed that X has an inverse; we can pre-multiply AX = XΛ by this inverse to get⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎣⎡===−−nλλλλLLMOMMMMOMMMLLLLLL00000000000032111ΛXΛXAXXAXX1−=Λ8Transform Example for 2 x 2• Last class example: eigenvectors for 2 x 2 matrix A, with λ1= 2 and λ2= 1⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡=052051)2()1(βααxxA• The X matrix and its inverse are⎥⎦⎤⎢⎣⎡αβα=05X⎥⎦⎤⎢⎣⎡αα−β−βα−α=−50))(()0)(5(11X9Check Inverse, Comupte AXIXXIXXX=⎥⎦⎤⎢⎣⎡+−+−−+−+−=⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣⎡−−−==⎥⎦⎤⎢⎣⎡−−−=−−−)0)(5())(())(5()5)(()0)(())(0())(()5)(0(105501?501111αβαααααββαβααβαβαααβαβααβαβDoes⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡++++=⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣⎡=0210)0)(2())(0())(2()5)(0()0)(5())(1())(5()5)(1(052051αβαβααβαααβαAX10Transform Example Result• This example produces the expected result: X-1AX is a diagonal matrix of eigenvalues (regardless of α and β)⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡−−−=⎥⎦⎤⎢⎣⎡+−+−−+−+=⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣⎡−−−=−2110010020021)0)(5())(()2)(5()10)(()0)(())(0()2)(()10)(0(0210501λλαβαβαβαβαααααββαβααβαααβαβAXX11Another Example• Last class 3 x 3 example had A matrix with eigenvalues λ1= -2, λ2= -2, λ3= 6 and eigenvectors shown below () ( ) ()⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−−−=112021203112312622321xxxA⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−−−⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−−−600020002102120213112312622102120213112Orthogonal Matrices• An orthogonal matrix has mutually orthogonal columns, a(j)• Write matrix as [a(1)a(2)a(3)… a(n)] •(a(i), a(j)) = aT(i)a(j)= Σakiakj= δij• Summation formula is equivalent to matrix multiplication of ATA = I• Thus, AT= A-1for orthogonal matrices• Both rows and columns are orthogonal313More on Orthogonal Matrices• A vector transform with an orthogonal matrix preserves the vector length•For y = Ax, with A orthogonal, ||y||2= yTy = (Ax)TAx = xTATAx• Orthogonal matrix: AT= A-1so ATA = I•So, ||y||2= xTATAx = xTIx = xTx =||x||2• Conclusion: when y = Ax, with Aorthogonal, ||y||2= ||x||214Hermitian/Symmetric Matrices• Symmetric matrix: A = AT• Hermitian matrix: AH= A†= (A*)T= A• A real symmetric matrix is a Hermitian matrix (also called self-adjoint)• For an n x n Hermitian matrix– Eigenvalues are real– Eigenvectors form a linearly independent, orthogonal basis set in n dimensions• May have complex eigenvectors for complex A15Unitary Matrix• Analog of an orthogonal matrix for complex-values matrices• For a unitary matrix, U, UH= U-1– I. e. for a unitary matrix we get the inverse by taking the transpose and setting all values of i to –i• Eigenvectors of a Hermitian matrix, A– Form an orthogonal matrix for real-valued A and a unitary matrix if A has complex values16Hermitian Eigenvectors•Recall X matrix whose columns are eigenvectors giving Λ = X-1AX• Requires X to have inverse• This is guaranteed for a Hermitian A• Furthermore, since X columns are orthogonal eigenvectors, X-1= XH, which is the same as XTfor real A⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=131302121A• Example of real Hermitian matrix, A17Hermitian Example• Solve Det[A – Iλ] = 0 for eigenvalues• λ1= 4.7131967, λ2= -2.789263462, and λ3= 0.076066756 •Solve (A – Iλk)x(k) =0 for unit eigenvectors and construct X matrix⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=553478.00.5528010.622955104950.00.788297-0.6062780.826225-0.2701830.494321X18Hermitian Example Continued• Show eigenvectors are orthonormal⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=553478.00.5528010.622955104950.00.788297-0.6062780.826225-0.2701830.494321X() ( )()()0 0.552801)0.622955)(()(-0.788297(0.606278)0.270183)0.494321)((,2)1(21=++== xxxxT• Can show (x(i) , x(j)) = δij419Hermitian Example Concluded• Since columns of X are orthogonal, X is an orthogonal matrix: X-1= XT⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=−553478.0104950.00.826225-0.5528010.788297-0.2701830.6229550.6062780.4943211X• Can verify this by taking inverse• Can also show that X-1AX = diag[λ1, λ2, λ3]20Complex Hermitian Example• Find X such that X-1AX = Λ for A = AH⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−=0010010iiA00101)(
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