Orthogonal Matrices Symmetric Matrices Hungyi Lee Outline Orthogonal
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Orthogonal Matrices & Symmetric Matrices Hung-yi Lee
Outline Orthogonal Matrices • Reference: Chapter 7. 5 Symmetric Matrices • Reference: Chapter 7. 6
Norm-preserving • A linear operator is norm-preserving if For all u Example: linear operator T on R 2 that rotates a vector by . Is T norm-preserving? Example: linear operator T is refection Is T norm-preserving?
Norm-preserving • A linear operator is norm-preserving if For all u Example: linear operator T is projection Is T norm-preserving? Example: linear operator U on Rn that has an eigenvalue ± 1. U is not norm-preserving, since for the corresponding eigenvector v, U(v) = v = · v v .
Orthogonal Matrix • An nxn matrix Q is called an orthogonal matrix (or simply orthogonal) if the columns of Q form an orthonormal basis for Rn • Orthogonal operator: standard matrix is an orthogonal matrix. unit is an orthogonal matrix. orthogonal
Norm-preserving • Necessary conditions: Normpreserving Orthogonal ? Matrix ? ? ? Linear operator Q is norm-preserving qj = 1 qi and qj are orthogonal qj = Qej = ej 畢式定理 qi + qj 2 = Qei + Qej 2 = Q(ei + ej) 2 = ei + ej 2 = qi 2 + qj 2
Orthogonal Matrix Those properties are used to check orthogonal matrix. • Simple inverse Q preserves dot projects Q preserves norms Normpreserving Orthogonal Matrix
Orthogonal Matrix • Proof (a) QQT = In det(In) = det(QQT) = det(Q)det(QT) = det(Q)2 det(Q) = ± 1. (b) (PQ)T = QTPT = Q 1 P 1 = (PQ) 1. Rows and columns
Orthogonal Operator • Preserves dot product Preserves norms
Example: Find an orthogonal operator T on R 3 such that Norm-preserving Also orthogonal
Conclusion • Orthogonal Matrix (Operator) • Columns and rows are orthogonal unit vectors • Preserving norms, dot products • Its inverse is equal its transpose
Outline Orthogonal Matrices • Reference: Chapter 7. 5 Symmetric Matrices • Reference: Chapter 7. 6
Eigenvalues are real • The eigenvalues for symmetric matrices are always real. Consider 2 x 2 symmetric matrices How about more general cases? 實係數多項式虛根共軛 The symmetric matrices always have real eigenvalues.
Orthogonal Eigenvectors Factorization A is symmetric Eigenvalue: …… Eigenspace: (dimension ) …… Independent orthogonal
Orthogonal Eigenvectors •
Diagonalization A is symmetric : simple P 560 ? P is an orthogonal matrix D is a diagonal matrix Diagonalization P consists of eigenvectors , D are eigenvalues
Diagonalization • Example A has eigenvalues 1 = 6 and 2 = 1, with corresponding eigenspaces E 1 = Span{[ 1 2 ]T} and E 2 = Span{[ 2 1 ]T} orthogonal T T B 1 = {[ 1 2 ] / 5} and B 2 = {[ 2 1 ] / 5}
Example of Diagonalization of Symmetric Matrix 1 = 2 2 = 8 Intendent Not orthogonal Gram. Schmidt normali zation normalization P is an orthogonal matrix
Diagonalization P is an orthogonal matrix A is symmetric P consists of eigenvectors , D are eigenvalues Finding an orthonormal basis consisting of eigenvectors of A (1) Compute all distinct eigenvalues 1, 2, , k of A. (2) Determine the corresponding eigenspaces E 1, E 2, , Ek. (3) Get an orthonormal basis B i for each Ei. (4) B = B 1 B 2 B k is an orthonormal basis for A.
Diagonalization of Symmetric Matrix Orthonormal basis simple Properly selected Eigenvectors form the good system A is symmetric Properly selected
Spectral Decomposition Orthonormal basis A = PDPT Let P = [ u 1 u 2 un ] and D = diag[ 1 2 n ]. = P[ 1 e 1 2 e 2 nen ]PT = [ 1 Pe 1 2 Pe 2 n. Pen ]PT = [ 1 u 1 2 u 2 nun ]PT
Spectral Decomposition A = PDPT Orthonormal basis Let P = [ u 1 u 2 un ] and D = diag[ 1 2 n ].
Spectral Decomposition • Example Find spectrum decomposition. Eigenvalues 1 = 5 and 2 = 5.
Conclusion • Any symmetric matrix • has only real eigenvalues • has orthogonal eigenvectors. • is always diagonalizable A is symmetric P is an orthogonal matrix
Appendix
Diagonalization • By induction on n. • n = 1 is obvious. • Assume it holds for n 1, and consider A R(n+1). • A has an eigenvector b 1 Rn+1 corresponding to a real eigenvalue , so an orthonormal basis B = {b 1, b 2, , bn+1} • by the Extension Theorem and Gram-Schmidt Process.
S = ST Rn n an orthogonal C Rn n and a diagonal L Rn n such that CTSC = L by the induction hypothesis.
Example: reflection operator T about a line L passing the origin. Question: Is T an orthogonal operator? (An easier) Question: Is T orthogonal if L is the x-axis? b 1 is a unit vector along L. b 2 is a unit vector perpendicular to L. P = [ b 1 b 2 ] is an orthogonal matrix. B = {b 1, b 2} is an orthonormal basis of R 2. [T]B = diag[1 1] is an orthogonal matrix. Let the standard matrix of T be Q. Then [T]B = P 1 QP, or Q = P[T]B P 1 Q is an orthogonal matrix. T is an orthogonal operator.
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