CS545_HW1

# CS545_HW1 - 1 Linear Algebra a 1 n = m y Ax A 1 y A 1 Ax if...

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1) Linear Algebra a) 1. n = m 1 1 1 1 1 if is a nonsingular matrix then det( ) 0 1 ( ) ( ) ( ) ( ) y Ax A y A Ax A A A A I A y Ix x A Adj A det A Adj A x y det A - - - - - = = = = = = ∴ = Q Q 1 A - is an inverse of matrix. If ( ) r A n = , x has only one solution. If ( ) r A n < , x has indefinite solutions. 2. n > m 1 1 1 1 1 , if ( ) then ( ) ( ) ( ) ( ) ( ) ( ) l l T T l T T T T T T T T n m y Ax A y A Ax r A m A A A A A A A y A A A Ax A A A A I x A A A y + - - - - - = = = = = = ∴ = Q Q 1 ( ) T T A A A - is a left pseudo-inverse of matrix. There is indefinite solution for x if ( ) r A m = which means there is more equations than unknowns in the matrix. 1 ( ) T T A A A - minimizes || || y Ax - . 3. n < m 1 1 1 1 1 , if ( ) then ( ) ( ) ( ) ( ) ( ( ) ) r r T T r T T T T T T T T n m y Ax A y AA x r A n A A AA A AA y AA AA x x A AA y I A AA A k + - - - - - < = = = = = ∴ = + - Q 1 ( ) T T A AA - is a right pseudo-inverse of matrix. 1

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CS545_HW1 - 1 Linear Algebra a 1 n = m y Ax A 1 y A 1 Ax if...

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