$$ eginpmatrix 1&-1 \ -1&2 \ -1&0 endpmatrix eginpmatrix x\y endpmatrix = eginpmatrix 3 \ 4 \ 5endpmatrix $$

So Assuming this goes by $ Avecx=vecb$

Then using.... $A^T A vecx = A^T vecb $

$$ eginpmatrix 1&-1&-1 \ -1&2&0 endpmatrix eginpmatrix 1&-1 \ -1&2 \ -1&0 endpmatrix vecx= eginpmatrix 1&-1&-1 \ -1&2&0 endpmatrix eginpmatrix 3 \ 4 \ 5endpmatrix $$

Which solves to be $$ eginpmatrix 3&-3 \ -3&5 endpmatrix vecx = eginpmatrix -6 \ 5 endpmatrix $$

which i solve to get $$ vecx = eginpmatrix -7/3 \ 3/2 endpmatrix $$


linear-algebra least-squares
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edited Mar 10 "17 at 1:38
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dantopa
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asked Nov 3 "15 at 4:00
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PatrickPatrick
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Because you are only asking for the solution...

You are watching: Find the least squares solution of the system

$$ left( eginvmatrix 1 & -1 \ -1 & 2 \ -1 & 0 endvmatrix^intercal eginvmatrix 1 & -1 \ -1 & 2 \ -1 & 0 endvmatrix ight) eginpmatrix x \ y endpmatrix = left( eginvmatrix 1 & -1 \ -1 & 2 \ -1 & 0 endvmatrix^intercal eginpmatrix 3 \ 4 \ 5 endpmatrix ight) $$

$$ eginvmatrix 3 & -3 \ -3 & 5 endvmatrix eginpmatrix x \ y endpmatrix = eginpmatrix -6 \ 5 endpmatrix $$

$$ eginpmatrix x = -frac52 \ y = -frac12 endpmatrix $$


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answered Nov 3 "15 at 4:37
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John AlexiouJohn Alexiou
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The normal equations are correct:$$lutz-heilmann.infobfA^*lutz-heilmann.infobfA x = lutz-heilmann.infobfA^*b.$$The error is in the solution which need to be$$ x = left( lutz-heilmann.infobfA^*lutz-heilmann.infobfA ight)^-1 lutz-heilmann.infobfA^*b.$$The inverse is$$left( lutz-heilmann.infobfA^*lutz-heilmann.infobfA ight)^-1 = frac1 6left(eginarraycc 5 & 3 \ 3 & 3 \endarray ight)$$which provides$$ x = -frac12left(eginarrayr 5 \ 1endarray ight).$$


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answered Mar 9 "17 at 15:35
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dantopadantopa
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