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lecture4 - EECS 275 Matrix Computation Ming-Hsuan Yang...

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EECS 275 Matrix Computation Ming-Hsuan Yang Electrical Engineering and Computer Science University of California at Merced Merced, CA 95344 h ttp://faculty.ucmerced.edu/mhyang Lecture 4 1 / 17
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Overview Basic definition: orthogonality, orthogonal projection, distance between subspaces, matrix inverse Matrix decomposition: singular value decomposition 2 / 17
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Reading Chapter 4 of Numerical Linear Algebra by Lloyd Trefethen and David Bau Chapters 2 and 3 of Matrix Computations by Gene Golub and Charles Van Loan Chapter 3 of Mathematical Modeling of Continuous Systems by Carlo Tomasi 3 / 17
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Matrix multiplication Let A = [ a 1 , . . . , a n ], A IR m × n , then A x = n X j =1 x j a j The output vector is a linear combination of matrix columns with coefficients given by the entries of x Example 1 2 3 3 2 1 1 2 3 = 1 1 3 + 2 2 2 + 3 3 1 = 1 3 + 4 4 + 9 3 = 14 10 1 2 3 4 5 6 7 8 9 0 1 0 = 1 2 5 8 = 2 5 8 4 / 17
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Orthogonality A set vectors { x 1 , . . . , x n } in IR m is orthogonal if x > i x j = 0 when i 6 = j , and orthonormal if x > i x j = δ ij Orthogonal vectors are maximally independent for they point in totally different directions Subspace: A collection of subspaces S 1 , . . . , S p in IR m is mutually orthogonal if x > y = 0 whenever x S i and y S j for i 6 = j The orthogonal complement of a subspace S IR m is S = { y IR m : y > x = 0 x S } It can be shown that ran( A ) = null( A > ) The vectors v 1 , . . . , v k form an orthonormal basis for a subspace S IR m if they are orthonormal and span S 5 / 17
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Orthogonality (cont’d) A matrix Q IR m × m is said to be orthogonal if Q > Q = I If Q = [ q 1 , . . . , q m ] is orthogonal, then the q i form an orthonormal basis for IR
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