Find a formula for the least squares solution of ax=b when the columns of a are orthonormal1 .

Let be a rectangular matrix with column vectors , i.e.



Then we have



and the product of the two is



Because the columns of are orthonormal, we have



which means reduces to an matrix with ones along the diagonal and zero everywhere else, i.e.



where denotes the identity matrix. This means the solution to is given by

Answerthe simplified expression isexplanationthe given expression is,let us deal with the bracket first.the two negatives in the parenthesis will become positive.the expression now becomes,we now expand the parenthesis to get,we group like terms to obtain,this simplifies to,