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Lecture Notes for CSE 275 - Matrix Computation with Yang at UC-Merced (UC Merced)

Notes Information

Material Type:Note 4
Professor:Yang
Class:CSE 275 - Matrix Computation
Subject:Computer Science & Engineering
University:University of California-Merced
Term:--
Keywords:
  • Best Approximation
  • Similarities
  • Eigenvectors
  • Interpretation
  • Recognition
  • Decomposition
  • Eigenvalues
  • Their (singular)
  • Homogeneous
  • Correlations
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CSE 275 Matrix Computation Ming-Hsuan Yang Electrical Engineering and Computer Science University of California at Merced Merced, CA 95344 http://faculty.ucmerced.edu/mhyang Lecture 4 1/14 Overview Basic de nition: orthogonality, orthogonal projection, distance between subspaces, matrix inverse Matrix decomposition 2/14 Reading Chapters 2 and 3 of Matrix Computations by Gene Golub and Charles Van Loan 3/14 Orthogonality A set vectors fx1;:::;xng in IRm is orthogonal if x>i xj = 0 when i 6= j, and orthonormal if x>i xj = ij Orthogonal vectors are maximally independent for they point in totally di erent directions Subspace: A collection of subspaces S1;:::;Sp in IRm is mutually orthogonal if x>y = 0 whenever x2Si and y2Sj for i 6= j The orthogonal complement of a subspace S IRm is S? =fy2IRm : y>x = 0 8x2Sg It can be shown that ran(A)? = null(A>) The vectors v1;:::;vk form an orthonormal basis for a subspace S IRm if they are orthonormal and span S 4/14 Orthogonality (cont'd) A...

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