Lect15-1 - Linear Algebra, Lecture 15 Constantin Teleman UC...

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Linear Algebra, Lecture 15Constantin TelemanUC Berkeley10 March 2011Constantin Teleman(Berkeley)Linear Algebra 1510 March 20111 / 11
Diagonalization and Inner productsThe Spectral Theorem for Real Symmetric matricesRecall that a square matrixAis diagonalizable if it can be written asSDS-1for an invertibleSand a diagonalD. The columns ofSare theneigenvectors ofA, and the diagonal entries ofDthe matching eigenvalues.QUESTION:When can be arrange forSto be aorthogonalmatrix?We are asking when we can choose anorthonormaleigenbasis forA.A necessary condition is thatAissymmetric,A=AT: ifS-1=ST(orthogonality),AT= (SDS-1)T= (S-1)TDTST=SDS-1=A.Theorem (Spectral theorem for real symmetric matrices)A real symmetric matrix is diagonalizable in an orthonormal eigenbasis.Partial proof (minus diagonalizability).We check thattwo eigenvectorsv1,2for distinct eigenvaluesλ1,2are:λ2hv1|v2i=hv1|λ2v2i=hv1|Av2i=hATv1|v2i=hAv1|v2i=λ1hv1|v2i;soλ16=λ2⇒ hv1|v2i= 0. We can then assemble an orthonormalA-eigenbasis by choosing orthonormal bases in each eigenspace.Constantin Teleman(Berkeley)Linear Algebra 1510 March 20112 / 11
Diagonalization and Inner productsThe Spectral Theorem for Normal matrices*Orthogonality of eigenvectors applies to a larger class of matrices, but it isonly the symmetric case that guarantees diagonalizability overR.Nevertheless, for real matrices that are diagonalizable overCbut notR,we had a nice structure theorem (thenormal form), which decomposedtheir action onRninto lines and planes. When is that decompositionorthogonal, and moreover, when do the plane actions involve genuinerotations and scalings, as opposed to ‘elliptic’ rotations?Theorem (Spectral theorem for normal matrices)The normal form A=SRS-1of a real matrix can be obtained with anorthogonal S iff A is anormalmatrix, that is, A commutes with AT.The proof, like the description of the real normal form, requires the use ofcomplex numbers and complex inner products; I will omit it.Normal matrices include the importantorthogonal matrices: ifQT=Q-1,thenQTQ=In=QQT. In particular, rigid motions of Euclidian space (in3 or more dims) take their normal form in an orthonormal basis.

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Term
Spring
Professor
Chorin
Tags
Differential Equations, Linear Algebra, Algebra, Equations, Matrices, Vector Space, Orthogonal matrix, Hilbert space, Constantin Teleman

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