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picture1_Matrix Pdf 174477 | Matrixtheory 20121011


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File: Matrix Pdf 174477 | Matrixtheory 20121011
Matrix Theory, Math6304 Lecture Notes from October 11, 2012 taken by Da Zheng 4 Variational characterization of eigenvalues, continued Werecall from last class that given a Hermitian matrix, we can ...

icon picture PDF Filetype PDF | Posted on 27 Jan 2023 | 2 years ago

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...Matrix theory math lecture notes from october taken by da zheng variational characterization of eigenvalues continued werecall last class that given a hermitian we can obtain its largest resp smallest eigenvalue maximizing minimizing the corresponding quadratic form over all unit vectors in fact due to following theorem courant and fischer any through min max or formula suppose mn is for each k n let s i denote set dimensional linear subspaces c also enumerate counting multiplicity increasing order e then have hax xi x kxk ii before starting proof an u which orhonormal as eigenvectors respectively first w span dimw so subspace should dim this because equality dims course p now choose note hx iu au it j follows pn ihu pnkxk here use thus sup indeed easy check since compact supremum attained sk on other hand consider particular pk hence choosing implies minimum conclude same idea shall omit similar details rst next therefore again gives nally remark compare result with horn johnson analy...
								

							

						
	
												
				
					
					
					
			
					
				
				
		
	

	
		





	

	
				

						

						
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Posted on: 27 Jan 2023 | 2 years ago
										
										
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