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01C SLAE LeastSquares RLS

# 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 a y 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 E.g. = . is an approximate solution of 2 3 2 3 x x i e 33 6 10 33 6 6 9 9 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 [ ] , . ., [ . ] . . E.g. = is the (unique) solution of E.g. = is also the (unique) solution of 2 3 E.g. is a (nonunique) solution of x x x x x x 0 0 0 3 2 6 3 6 9 3 4 2 3 18 = = = L N M O Q P = L N M O Q P = L N M O Q P = [ ][ ] [ ] [ ] [ ][ ] [ ] x = x x a x a x a x y a x a x a x y a x a x a x y n n n n m m mn n m 11 1 12 2 1 1 21 1 22 2 2 2 1 1 2 2 + + + = + + + = + + + = L L M L a a a a a a a a a x x x y y y n n m m mn n m 11 12 1 21 22 2 1 2 1 2 1 2 L L M O 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 a a a a x x y y n m mn n m 11 1 1 1 1 L M O M L M M

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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 1 2 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 a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a + = = = = = = + − + = = A A 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 = 2 6 3 9 is rank[ ] = since and det[2] = 1 E.g. The rank of = 2 6 3 10 is rank[ ] = since A A A A A A L N M O Q P = = = L N M O Q P = = = r r 1 2 6 3 9 2 9 6 3 0 2 2 6 3 10 2 10 6 3 2 0 , * * , * * E.g. The rank of = 6 20 3 9 30 = since there is only 1 independent column 2 3 or 6 9 or 20 30 E.g. The rank of = 2 6 6 3 9 10 since there are 2 independent columns 2 3 and 6 10 E.g. The rank of = 2 4 6 3 6 9 E.g. The rank of = 1 2 3 2 6 12 , 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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01C SLAE LeastSquares RLS - Ka C Cheok SIMULTANEOUS LINEAR...

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