pseudo inverse of a matrix

Hi

Let A be an $\displaystyle m \times n$ matrix. An $\displaystyle n\times m $ matrix $\displaystyle A{^\oplus}$ is a pseudoinverse of A if there are matrices $\displaystyle U \ and\ V$ such that

$\displaystyle AA^{\oplus} A= A,$

$\displaystyle A^{\oplus}=UA^{\oplus}=A^{T}V.$

Show that $\displaystyle A^{\oplus}$ exists and is unique.

Whereas i know pseudo inverse is obtained from singular value decomposition (SVD), as any $\displaystyle m \times n$ matrix can be decomposed as

$\displaystyle A=UDV^T$

where $\displaystyle D$ is diagonal matrix and $\displaystyle U, V$ are orthogonal matrices.

So we can find$\displaystyle A^{\oplus}=UD^{\oplus}V^T,$ this stratifies the first property $\displaystyle AA^{\oplus}A = A$.

Where $\displaystyle D^{\oplus} $ is matrix of reciprocal of eigenvalues of $\displaystyle D$.

My Question is how can

1. How can I prove other properties $\displaystyle A^{\oplus}=UA^{\oplus}=A^{T}V$ using this SVD decomposition.

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

Thanks in advance