Orthogonal Matrices Symmetric Matrices Hungyi Lee Outline Orthogonal

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Orthogonal Matrices & Symmetric Matrices Hung-yi Lee

Orthogonal Matrices & Symmetric Matrices Hung-yi Lee

Outline Orthogonal Matrices • Reference: Chapter 7. 5 Symmetric Matrices • Reference: Chapter 7.

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

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

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 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

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

Orthogonal Matrix Those properties are used to check orthogonal matrix. • Simple inverse Q preserves dot projects Q preserves norms Normpreserving Orthogonal Matrix

Orthogonal Matrix • Rows and columns Proof (a) QQT = In 1 = det(In)

Orthogonal Matrix • Rows and columns Proof (a) QQT = In 1 = det(In) = det(QQT) = det(Q)det(QT) = det(Q)2 det(Q) = ± 1. (b) (PQ)T = QTPT = Q 1 P 1 = (PQ) 1.

Orthogonal Operator • Preserves dot product Preserves norms

Orthogonal Operator • Preserves dot product Preserves norms

Example: Find an orthogonal operator T on R 3 such that Norm-preserving Also orthogonal

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 •

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.

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

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 Factorization A is symmetric Eigenvalue: …… Eigenspace: (dimension ) …… Independent orthogonal

Orthogonal Eigenvectors •

Orthogonal Eigenvectors •

Diagonalization A is symmetric : simple P 560 ? P is an orthogonal matrix

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

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

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 ,

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

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

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

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 =

Spectral Decomposition • Example Find spectrum decomposition. Eigenvalues 1 = 5 and 2 = 5.

Conclusion • Any symmetric matrix • has only real eigenvalues • has orthogonal eigenvectors.

Conclusion • Any symmetric matrix • has only real eigenvalues • has orthogonal eigenvectors. • is always diagonalizable A is symmetric P is an orthogonal matrix

Appendix

Appendix

Diagonalization • By induction on n. • n = 1 is obvious. • Assume

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

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

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.