Exploring Orthogonal and Symmetric Matrices - Insights by Hung-yi Lee
Unlock the intricacies of orthogonal and symmetric matrices with enlightening insights by Hung-yi Lee, diving into norm-preservation, necessary conditions, properties, and more in linear algebra.
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Presentation Transcript
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 R2that rotates a vector by . Is T norm-preserving? ? =1 0 Example: linear operator T is refection Is T norm-preserving? 0 1
Norm-preserving A linear operator is norm-preserving if ? ? = ? For all u Example: linear operator T is projection Is T norm-preserving? ? =1 0 0 0 Example: linear operator U on Rnthat 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 unit is an orthogonal matrix. orthogonal
Norm-preserving Necessary conditions: Norm- preserving Orthogonal Matrix ? ??? Linear operator Q is norm-preserving qj = 1 qj = Qej = ej qiand qjare orthogonal qi+ qj 2= Qei+ Qej 2= Q(ei+ ej) 2 = ei+ ej 2= 2 = qi 2+ qj 2
Those properties are used to check orthogonal matrix. Orthogonal Matrix Q is an orthogonal matrix ???= ?? ? is invertible, and ? 1= ?? ?? ?? = ? ? for any u and v ?? = ? for any u Simple inverse Q preserves dot projects Q preserves norms Norm- preserving Orthogonal Matrix
Orthogonal Matrix Let P and Q be n x n orthogonal matrices ???? = 1 ?? is an orthogonal matrix ? 1 is an orthogonal matrix ?? is an orthogonal matrix Check by ?? 1= ??? Check by ? 1 1= ? 1 ? Proof (a) QQT= In det(In) = det(QQT) = det(Q)det(QT) = det(Q)2 det(Q) = 1. (b) (PQ)T = QTPT= Q 1P 1 = (PQ) 1. Rows and columns
Orthogonal Operator Applying the properties of orthogonal matrices on orthogonal operators T is an orthogonal operator ? ? ? ? = ? ? for all ? and ? ? ? = ? for all ? Preserves dot product Preserves norms T and U are orthogonal operators, then ?? and ? 1 are orthogonal operators.
Example: Find an orthogonal operator T on R3such that 0 1 0 1 2 Norm-preserving ? = 0 1 2 1 2 Find ? 1 first Because ? 1= ?? ? = ? 1?2 ?? = ?2 ? = 0 1 2 Also orthogonal 1 2 ? 1= 0 1 2 1 2 0 1 0 1 2 ? 1= 0 0 1 2 1 2 1 2 0 1 0 1 1 2 2 0 0 1 1 1 2 0 ? = ? 1 ?= 2 1 2 0 0
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. How about more general cases? Consider 2 x 2 symmetric matrices ??? ? ??2 = ?2 ? + ? ? + ?? ?2 The symmetric matrices always have real eigenvalues.
Orthogonal Eigenvectors A is symmetric ??? ? ??? Factorization ?1? ?2 ?2 ? ?? ?? = ? ?1 ?1 ?1 ?2 ?2 ?? Eigenvalue: ?? ?? ?2 ?1 Eigenspace: (dimension) orthogonal Independent
Orthogonal Eigenvectors A is symmetric. If ? and ? are eigenvectors corresponding to eigenvalues ? and ? (? ?) ? and ? are orthogonal.
Diagonalization A = ?DP? P560 ? A is P?A? = D symmetric P is an orthogonal matrix D is a diagonal matrix : simple P?A? = D P 1A? = D A = ?DP 1 Diagonalization A = ?DP? P consists of eigenvectors , D are eigenvalues
Diagonalization Example A = ?DP? 2 2 A = ?DP 1 = A 2 5 P?A? = D A has eigenvalues 1= 6 and 2= 1, with corresponding eigenspaces E1 = Span{[ 1 2 ]T} and E2 = Span{[ 2 1 ]T} orthogonal B1 = {[ 1 2 ]T/ 5} and B2 = {[ 2 1 ]T/ 5} 1 2 2 6 0 0 1 1 = = and . P D 1 5
Example of Diagonalization of Symmetric Matrix A = ?DP? A = ?DP 1 P is an orthogonal matrix Gram- Schmidt Intendent 1= 2 1 1 6 6 1 1 0 1 0 1 1 1 2 Eigenspace: ???? , ???? , 2 normali zation 0 2 6 Not orthogonal 2= 8 1 1 1 3 3 3 1 1 1 Eigenspace: ???? ???? normalization 1 1 6 6 1 1 1 3 3 3 2 0 0 0 2 0 0 0 8 1 1 2 ? = ? = 2 0 2 6
Diagonalization P is an orthogonal matrix A is P?A? = D symmetric A = ?DP? P consists of eigenvectors , D are eigenvalues Finding an orthonormal basis consisting of eigenvectors of A (1) Compute all distinct eigenvalues 1, 2, , kof A. (2) Determine the corresponding eigenspaces E1, E2, , Ek. (3) Get an orthonormal basis Bifor each Ei. (4) B = B 1 B 2 B kis an orthonormal basis for A.
Diagonalization of Symmetric Matrix ? = ?1?1+ ?2?2+ + ???? ? ?1 ? ?2 ? ?? Orthonormal basis ?B ? ?B ? ?B simple Eigenvectors form the good system ? 1 ? 1 ? ? Properly selected Properly selected ? = ??? 1 ? ? ? A is symmetric
Spectral Decomposition Orthonormal basis Let P = [ u1u2 un] and D = diag[ 1 2 n]. A = PDPT = P[ 1e1 2e2 nen]PT = [ 1Pe1 2Pe2 nPen]PT= [ 1u1 2u2 nun]PT ?1 ?2 ?? = ?1P1+ ?2P2+ + ??P? ?? are symmetric
Spectral Decomposition Orthonormal basis A = PDPT Let P = [ u1u2 un] and D = diag[ 1 2 n]. = ?1P1+ ?2P2+ + ??P?
Spectral Decomposition Example 3 4 3 ? = Find spectrum decomposition. 4 4 5 2 5 1 5 Eigenvalues 1= 5 and 2= 5. ?= ?1= ?1?1 2 5 1 5 2 5 An orthonormal basis consisting of eigenvectors of A is 2 1 5 ?1 2 5 4 5 ?= ?2= ?2?2 ?2 5 1 2 5 5 ? = , ? = ?1?1+ ?2?2
Conclusion Any symmetric matrix has only real eigenvalues has orthogonal eigenvectors. is always diagonalizable P?A? = D A = ?DP? A is symmetric P is an orthogonal matrix
Diagonalization By induction on n. n = 1 is obvious. Assume it holds for n 1, and consider A R(n+1) (n+1). A has an eigenvector b1 Rn+1corresponding to a real eigenvalue , so an orthonormal basis B = {b1, b2, , bn+1} by the Extension Theorem Extension Theorem and Gram- Schmidt Process.
T T T T b b b b b b b A A A + 1 T 1 T 1 1 T 2 1 T 1 n b b b b b b b A A A + = = T 2 2 1 2 2 2 1 n b b b B AB A A A + 1 2 1 n T n T n T n T n b b b b b b b A A A + + + + + 1 0 1 1 1 2 1 1 n T = = = and = = T T T j T b b b b b b b b since , 0 . 1 A A A j 1 1 1 1 1 1 j 0 S S = ST Rn n an orthogonal C Rn nand a diagonal L Rn n such that CTSC = L by the induction hypothesis. T T T T T T T 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 = = = T B A B T T T 0 0 0 C C C S C C SC L orthogonal P orthogonal P diagonal D 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? b1is a unit vector along L. b2is a unit vector perpendicular to L. P = [ b1b2] is an orthogonal matrix. B = {b1, b2} is an orthonormal basis of R2. [T]B= diag[1 1] is an orthogonal matrix. Let the standard matrix of T be Q. Then [T]B= P 1QP, or Q = P[T]BP 1 Q is an orthogonal matrix. T is an orthogonal operator.
