Orthogonal and Symmetric Matrices
In this informative content, Hung-yi Lee discusses the concepts of orthogonal and symmetric matrices, exploring their properties, applications, and relationships with linear operators. Learn about norm-preservation, necessary conditions for norm-preserving matrices, and practical methods to check if a matrix is orthogonal. Dive into the world of linear algebra with a focus on orthogonal and symmetric matrices.
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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
Orthogonal Matrix An nxn matrix Q is called an orthogonal matrix if the columns of Q are orthonormal. Orthogonal operator: standard matrix is an orthogonal matrix. unit unit is an orthogonal matrix. orthogonal
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 reflection 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 .
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 1 ?? ??= ????? Q is an orthogonal matrix ??? = ?? ? is invertible, and ? 1= ?? ?? ?? = ? ? for any u and v ?? = ? for any u 1 ? ?: 0 ? = ?: 1 2 ??? i-j entry is ????? 3 ?? ?? = ?????= ?????? = ???? = ??? = ? ? 2 ?? ?? = ? ? ?? ?? = ? ? 3 2= ? 2 ?? ?? = ?
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. (C) (Q-1)-1 = (QT)-1= (Q-1)T 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 0 1 0 1 2 0 1 1 2 2 0 0 1 1 1 2 ? = ? 1 ?= 1 2 2 0 0 2?1+ 0?2+ 1 1 2?3= 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. Consider 2 x 2 symmetric matrices How about more general cases? ??? ? ??2 = ?2 ? + ? ? + ?? ?2 The symmetric matrices always have real eigenvalues.
Symmetric matrix A always has eigenvalue. Eigenvalues are real A symmetric matrix A has an eigenvalue ? ? = ? + ?? ? ? = ? ? ? ? = ? ? Av = ?? ?? = ?? ? = ? ?T? ?T?? = ? ?1+ ?1? ?2+ ?2? ?1 ?1? ?2 ?2? ? = ? ?T?? ?T?T? = ? ?T? T? = ? ? = ?T? = ? ? ? = ? 2+ ?T? =? 2+ ?1 ? = 0 > 0 2+ ?2 2+ ?2 = ?1
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? 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
A is P?A? = D Diagonalization symmetric ?:? ? ? has eigenvalue ? ??1= ??1 ?1 is unit vector Find an orthonormal basis ?1,?2, ,?? = ? eigenvector don t care by the Extension Theorem and Gram-Schmidt Process ???? =? ?????= ?????? ?= ???? symmetric ?????1 = ????1 = ????1 = ????1 ?1? ?2? ? 0 1 0 ? 0 0 A = ? ?1 = ? = symmetric
A is P?A? = D Diagonalization symmetric ?:? ? ortho ? ? 0 0 1 0 ???? = ? ?? ? = ? = 0 A 0 A 0 ? ortho sym symmetric symmetric ortho ??????? =? ? 0 1 ? 1 0 0 0 = ? ?? ? ? ? 0 0 0 0 A ?
A is P?A? = D Diagonalization symmetric ?:? ? ??????? =? ? ?? ? = ? 1 ? 1 0 0 0 0 ? ? 0 0 0 0 A A ? ortho ortho ? ? 0 0 ? 0 = = ??????? ? ?? ? 0 0 0 A = ?
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 independent 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 nx1 1xn ?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
