The SVD of the matrix $A\in \mathbb{C^{M\times N}}$ is $A=U\Sigma V^H$ where $U \in \mathbb{C}^{M\times M}, V \in \mathbb{C}^{N\times N}$ and $\Sigma$ is diagonal with the diagonal elements as the singular values ($\sigma_1^2, \sigma_2^2, ..., \sigma_m^2$). Let $P$ be a linear transformation constructed by having the last $N-1$ columns of $V$ as its columns. Since the columns of $V$ represents an orthonormal basis, the multiplication $AP$ has the effect of zeroing the first singular value of $A$.
My goal is to zero the first singular value $\sigma_1^2$ and weight the next two ($\sigma_2^2, \sigma_3^2$) by factors $\alpha_1$ and $\alpha_2$. I can achieve that by multiplying the first two columns of $P$ by $\alpha_1$ and $\alpha_2$. Therefore the multiplication $AP$ has the effect of weighting the second and third singular values, i.e, $\alpha_1 \sigma_2^2, \alpha_2 \sigma_3^2$. However $P$ no longer has orthonormal columns since the first two columns are weighted by $\alpha_1$ and $\alpha_2$ . I'm looking for a transformation $P$ that has the same effect of weighting the singular values, but with orthonormal columns. Does this kind of transformation exist. Secondly, is there any square matrix that can achieve the same result, it has to be non-orthogonal as square orthogonal matrices don't change the frobenius norm.