pseudo inverse of a matrix

Hi

Let A be an matrix. An matrix is a pseudoinverse of A if there are matrices such that

Show that exists and is unique.

Whereas i know pseudo inverse is obtained from singular value decomposition (SVD), as any matrix can be decomposed as

where is diagonal matrix and are orthogonal matrices.

So we can find this stratifies the first property .

Where is matrix of reciprocal of eigenvalues of .

My Question is how can

1. How can I prove other properties using this SVD decomposition.

2. Here in the stated problem matrices are not orthogonal, is it OK.

Thanks in advance