01C SLAE LeastSquares RLS - Ka C. Cheok SIMULTANEOUS LINEAR...

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Ka C. Cheok 1 SLAE & LeastSquares 09/03/05 1 S IMULTANEOUS L INEAR A LGEBRAIC E QUATIONS (SLAE) Statement of problem: Consider a set of m simultaneous linear algebraic equations (SLAE) with n unknown varia bles or where ay ij i and are known or given values and x i are the unknown variables to be solved. The SLAE can be short handed as The problem is to find x, given the equation Ax y = , with A and y specified. Definition of Solutions of SLAE 1. x = x 0 is defined as a solution of Ax=y , only if Ax 0 = y. 2. If no such x 0 can be found then we say that Ax = y is inconsistent and has no exact solution . 3. If there is an x 0 such that Ax y 0 , then we say that x 0 is an approximate solution . E.g. One cannot find an x such that 2 3 L N M O Q P = L N M O Q P [] x 6 10 = . is an approximate solution of 2 3 2 3 xx i e 33 6 10 66 99 6 10 L N M O Q P = L N M O Q P L N M O Q P = L N M O Q P L N M O Q P ,.. , [.] . . = is the (unique) solution of = is also the (unique) solution of 2 3 is a (nonunique) solution of x x 0 0 0 32 6 3 6 9 3 4 23 1 8 = = = L N M O Q P = L N M O Q P = L N M O Q P = [] [] [ ] [ ] x=x x ax ax ax y nn mm m n n m 11 1 12 2 1 1 21 1 22 2 2 2 11 22 ++ += L L M L aa a a a x x x y y y n n m n n m 11 12 1 21 22 2 12 1 2 1 2 L L MO L M M L N M M M M O Q P P P P L N M M M M O Q P P P P = L N M M M M O Q P P P P Ax y A x y == L N M M M O Q P P P = L N M M M O Q P P P = L N M M M O Q P P P where x x y y n n n m 11 1 1 L MOM L MM
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Ka C. Cheok 1 SLAE & LeastSquares 09/03/05 2 Determinant of a Matrix () 11 12 11 12 11 22 12 21 21 22 21 22 11 12 13 12 22 23 21 23 21 22 21 22 23 11 12 13 32 33 31 33 31 32 31 32 33 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 det det 1 aa aaa a a a aaaa +  = =−= =   == + + AA A A your turn Singular and Nonsigular Matrix Matrix A is said to be singular if 0 = A , and nonsingular if 0 A . Determining the Rank of a Matrix 1. The rank of matrix A , denoted as rank[ A ] or r A , is defined as the size of the largest nonsingular minor square matrix that can be formed from A . 2. A matrix rank is the smaller of the number of independent rows or column in the matrix. 3. The maximal rank of an m x n matrix is the smaller of m and n, i.e., ) , min( n m r A E.g. The rank of = 26 39 is rank[ ] = since and det[2] = 1 The rank of = 31 0 is rank[ ] = since A A L N M O Q P = L N M O Q P = r r 1 29 630 2 0 21 0 63 2 0 ,* * * The rank of = 62 0 393 0 = since there is only 1 independent column 2 3 or 6 9 or 20 30 The rank of = 26 6 391 0 since there are 2 independent columns 2 3 and 6 10 The rank of = 246 369 The rank of = 12 3 261 2 , 3 independent columns, 2 independent rows Similar arguments apply when has more rows than columns. A A A A A 2 1 2 L N M O Q P L N M O Q P L N M O Q P L N M O Q P L N M O Q P = L N M O Q P L N M O Q P L N M O Q P = L N M O Q P = , , __?
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Ka C. Cheok 1 SLAE & LeastSquares 09/03/05 3 4. If A is m x n and m n , then A has full rank m if det[ AA’ ] > 0. 5. If A is m x n and m n , then A has full rank n if det[ A’A ] > 0.
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This note was uploaded on 04/17/2011 for the course SYS 635 taught by Professor Re during the Spring '11 term at Albany College of Pharmacy and Health Sciences.

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01C SLAE LeastSquares RLS - Ka C. Cheok SIMULTANEOUS LINEAR...

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