5. (5 points, bonus). Find the Singular Value Decomposition of the matrix A from problem 4. Verify your answer.

The question is not complete.

I will explain the concept of Singular Value Decomposition

Step-by-step explanation:

The singular value decomposition (SVD) of a matrix A is the process of factorizing A into the  product of three matrices A = where the columns of U and V are orthonormal and  the matrix D is diagonal with positive real entries.

This process is used to carry out many tasks.

Data matrix A is useful to find a low rank matrix which is a good approximation  to the data matrix.

The columns of V is called the right singular  vectors of A and it normally form an orthogonal set with no assumptions on A. The columns  of U is reffered to as the left singular vectors and it also forms an orthogonal set. The orthogonality forms the inverse of A for a square and invertible matrix A which is .

If we want gain better insight of the process, we will need to treat the rows of an n × d matrix A as n points in a  d-dimensional space and find the best k-dimensional subspace  with respect to the set of points. This means that we will minimize the sum of the squares of the  perpendicular distances of the points to the subspace.

To find the best fitting line through the origin with respect  to a set of points {xi  |1 ≤ i ≤ n} will mean that we will minimize the sum of the squared  distances of the points to the line.