Given A=( 1 2 -1 0 -2 3 1 5 -1 -3 1 1 ) an

Given A=( 1 2 -1 0 -2 3 1 ...

Key Concepts

Residual (Error Vector)

The residual, or error vector, is the difference between the observed vector b and the projected vector Ax (i.e., the least squares approximation). The norm of this residual quantifies the overall error in the approximation. A smaller norm indicates a closer fit, while the residual itself provides insight into how much of b lies outside the column space of A.

Orthogonal Projection

The concept of orthogonal projection is central to least squares analysis. The least squares solution can be interpreted as the projection of the vector b onto the column space of the matrix A. The projected vector is the closest point within that subspace to the original vector b, ensuring that the error (residual) is orthogonal to the space spanned by A’s columns.

Normal Equations

Normal equations arise from setting the gradient of the least squares cost function (the sum of squared residuals) to zero. The resulting equation, A?Ax = A?b, provides a way to compute the least squares solution. It effectively transforms the problem into solving a system of equations where the solution minimizes the error in the Euclidean norm.

Least Squares Method

The least squares method is a mathematical approach used to find an approximate solution to an overdetermined system by minimizing the sum of the squares of the residuals—the differences between the observed values and those predicted by the model. This method is essential in data fitting, regression analysis, and many applied fields where exact solutions are unattainable.

Overdetermined System

An overdetermined system is one in which there are more equations than unknowns. In such systems, it is often impossible to find an exact solution that satisfies every equation simultaneously. Instead, the goal is to find an approximate solution that minimizes the deviation from all the equations as much as possible.

Transcript

00:02 In this question, we are asked to find the least squares solution of the system corresponding to this matrix and this vector.

00:12 That is, a system ax equals b, where x is our vector of four, or sorry, of three variables.

00:24 So first of all, what is the least squares solution? it is the solution to this inconsistent system that gets the closest to, it's the closest vector to an actual solution, the closest we can get, because it minimizes the sum of the squares of the components of the vector b minus ax.

00:58 So the formula to find this matrix, or rather the equation to find this solution, is a transpose ax equals a transpose b.

01:15 And if a transpose a is invertible, a transpose a inverse times a transpose b.

01:30 In our case, we'll have to find a transpose a and a transpose b, invert a transpose a, and then multiply them.

01:46 So first of all, a transpose a.

01:53 The diagonal entries are easy to find in this case, and in any case when we have a matrix times its transpose, or rather the transpose times the matrix, because the order of multiplication matters.

02:11 So in these cases, the diagonal entries are given by the sums of the squares of the appropriate column or row, so column of the original matrix.

02:23 The first diagonal entry thus is one squared plus two squared plus one squared plus one squared, also known as seven.

02:34 The middle entry is the sum of the squares of the middle column, so that's one squared plus one squared plus one squared, or three.

02:45 And finally, the last column is four plus one plus one.

03:01 Now for the other entries, we combine rows and columns.

03:12 So for the first row, second column, we take the dot product of the first row of the transposed matrix and the first column of a...

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