Solution
= Solution
Using the <orthonormal> columns of $U_1,U_2$ in the factorization gives
$$
I_n=A^TA=V(\Sigma_1^2+\Sigma_2^2)V^T.
$$
Multiplying by the <orthogonal matrices> $V^T$ and $V$ gives
$$
\boxed{\Sigma_1^2+\Sigma_2^2=I_n}.
$$
= Solution
Using the <orthonormal> columns of $U_1,U_2$ in the factorization gives
$$
I_n=A^TA=V(\Sigma_1^2+\Sigma_2^2)V^T.
$$
Multiplying by the <orthogonal matrices> $V^T$ and $V$ gives
$$
\boxed{\Sigma_1^2+\Sigma_2^2=I_n}.
$$