How Do You Calculate the Pseudoinverse of a Singular Matrix?

[ahamdiheme]

Homework Statement

To determine the pseudoinverse of this matrix.
1 1
2 2

Homework Equations

I find it difficult writing out the equation properly so i have attached it as a doc file.

The Attempt at a Solution

I have taken the limit as [tex]\delta[/tex] tends to zero, but getting the inverse of the matrix is impossible. I need to use this to find the pseudoinverse. Thank you

 

[mighty2000]

for any help in advance!

Thank you for your question. The pseudoinverse of a matrix is a generalization of the inverse for non-square matrices. It is denoted as A^+ and can be calculated using the singular value decomposition (SVD) of the matrix A.

The SVD of a matrix A is given by A = UΣV^T, where U and V are orthogonal matrices and Σ is a diagonal matrix containing the singular values of A. The pseudoinverse can then be calculated as A^+ = VΣ^+U^T, where Σ^+ is the pseudoinverse of Σ.

In your case, the matrix A = [1 1; 2 2] is not full rank, meaning it has linearly dependent columns. Therefore, it does not have an inverse. However, we can still calculate its pseudoinverse using the SVD. The SVD of A is A = [1 0; 0 0]Σ[1 1; 1 1] = UΣV^T, where Σ = [1 0; 0 0] and U = V = [1 0; 0 1]. The pseudoinverse is then A^+ = VΣ^+U^T = [1 0; 0 0]^+[1 0; 0 1]^T = [1 0; 0 0].

I hope this helps with your calculations. Let me know if you have any further questions.