Question

Calculate the SVD of matrix A =

2	2
-1	1

by hand and find the rank 1 approximation of A

Answer 1

We know that the Singular value decomposition of a matrix is to transform a given matrix A in the form ,

where U and V are orthogonal matrices and S is an m n diagonal matrix with diagonal elements .

Given matrix is -

Its Transpose matrix ,

Therefore ,

NOW WE WILL FIND THE EIGENVECTOR FOR THE ABOVE MATRIX (A.A'),

Thus ,

i.e., the eigenvalues of the matrix A.A' are 2,8

Therefore ,

• Eigenvector for = 8,

  

Length =

Since , U and V are orthogonal matrices , therefore U can be found out by normalising v , which is basically done in a manner as we find the normal vector of a given vector by dividing each elements with the Length of the matrix V .

So , on normalising it gives ,

• Eigenvector for = 2 ,

Length =

Therefore , on normalising , it gives ,

Now ,

and Since ,

Further , we have -

In ,

Rank - 1 approximation of the given matrix A is given as,

We only take the term which corresponds to the largest singular value , which is for i.e., 2.8284

Therefore , the rank - 1 approximation of A is given by ,

i.e.,

Hence , this is the best rank - 1 approximation of the SVD .