Some useful linear algebra-2

Some useful linear algebra-2 - Some useful linear algebra...

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Unformatted text preview: Some useful linear algebra Linearly independent vectors span(V): span of vector space V is all linear combinations of vectors v i, i.e. for only 2 1 3 3 2 2 1 1 = = = = = + + + + i i i v v v v i i i v singular is ) ( hence ) ( A I x A I x Ax- =- = The eigenvalues of A are the roots of the characteristic equation ) det( ) ( =- = A I p = =- N AS S . 2 1 1 Eigenvectors of A are columns of S diagonal form of matrix AM M B 1 If- = Similarity transform then A and B have the same eigenvalues The eigenvector x of A corresponds to the eigenvector M-1 x of B Rank and Nullspace 1 1 = m n n m b x A Least Squares b Ax = More equations than unknowns Look for solution which minimizes ||Ax-b|| = (Ax-b) T (Ax-b) Solve Same as the solution to LS solution ) ( ) ( = -- i T x b Ax b Ax b A Ax A T T = b A A A x T T 1 ) (- = Properties of SVD...
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This note was uploaded on 01/12/2012 for the course CMSC 733 taught by Professor Staff during the Spring '08 term at Maryland.

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Some useful linear algebra-2 - Some useful linear algebra...

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