# Singular values of block matrix and stacked block column matrix

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: $$$$\sigma_1(M_1) \leq \sigma_1(M_2)$$$$ where $$\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!

• Can you elaborate on your motivation for suspecting this, and on what your have tried so far? – Brian Jan 26 '20 at 22:37

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 $$A=B=\pmatrix{0&1\\ 0&1},\ C=D=\pmatrix{1&1\\ 1&1}, \ M_1=\pmatrix{0&1&1&1\\ 0&1&1&1\\ 0&1&1&1\\ 0&1&1&1\\}, \ M_2=\pmatrix{0&1\\ 0&1\\ 0&1\\ 0&1\\ 1&1\\ 1&1\\ 1&1\\ 1&1}.$$
• @Alex Sorry, no, except the really obvious ones (degenerate cases like $A=B=0$ or $C=D=0$, or trivial cases such as $M_1$ is unitary). – user1551 Jan 27 '20 at 20:26