Let $A, B, C, D \in \mathbb{R}^{n \times n}$, let $$ M_1 = \begin{bmatrix}A & C \\ B & D\end{bmatrix} \quad M_2 = \begin{bmatrix}A \\ B \\ C \\ D\end{bmatrix} $$ I suspect that: \begin{equation} \sigma_1(M_1) \leq \sigma_1(M_2) \end{equation} where $\sigma_1(M_1)$ is the maximum singular value of $M_1$.
We can show that: \begin{align} \sigma_1(M_2)^2 = \lambda_1(M_2^TM_2) &= \lambda_1(A^TA + B^TB + C^TC + D^TD) \\ \Leftrightarrow \sigma_1(M_2) &= \sqrt{ \lambda_1(A^TA + B^TB + C^TC + D^TD) } \end{align}
This equality for $M_1$ is false and numerical test suggest that $\sigma_1(M_1) \leq \sigma_1(M_2)$ and I haven't been able to find a counter example.
I am having trouble proving it. Thanks!
This isn't true. Random counterexample: we have $\|M_1\|_2=2\sqrt{3}=3.46>3.24=1+\sqrt{5}=\|M_2\|_2$ when $$ A=B=\pmatrix{0&1\\ 0&1},\ C=D=\pmatrix{1&1\\ 1&1}, \ M_1=\pmatrix{0&1&1&1\\ 0&1&1&1\\ 0&1&1&1\\ 0&1&1&1\\}, \ M_2=\pmatrix{0&1\\ 0&1\\ 0&1\\ 0&1\\ 1&1\\ 1&1\\ 1&1\\ 1&1}. $$