How to show that $\mathbf{A}\approx\mathbf{B}$

Question

I have two $M\times N$ matrices $\mathbf{A}=\mathbf{U_A^{}\Sigma_A^{}V_A^T}$ and $\mathbf{B}=\mathbf{U_B^{}\Sigma_B^{}V_B^T}$. I want to show that $\mathbf{A}\approx\mathbf{B}$ in every sense i-e

1. $\mathbf{U_A^{}}\approx\mathbf{U_B^{}}$
2. $\mathbf{V_A^{}}\approx\mathbf{V_B^{}}$
3. $\mathbf{\Sigma_A^{}}\approx\mathbf{\Sigma_B^{}}$

To prove $\mathbf{\Sigma_A^{}}\approx\mathbf{\Sigma_B^{}}$, I can use frobenius norm $\|\mathbf{X}\|_F=\sqrt{trace(\mathbf{X}^T\mathbf{X}^{})}=\sqrt{\sum_{i=1}^{r}\sigma^2_i}$.

But how do I prove 1 & 2.

One option is to show that $\mathbf{x^T(A-B)x}\leq \delta \quad \forall \quad \mathbf{x\neq0}$, where $0<\delta<<1$.