= Solution
Let $x_a\geq0$ be the <eigenvalues> of the positive semidefinite matrix $M^\dagger M$. The potential is
$$
V=\sum_{a=1}^N\left(kx_a+\frac\lambda2x_a^2\right).
$$
For $k<0$ and $\lambda>0$, each summand is minimized at
$$
x_a=v^2=-\frac{k}{\lambda},
$$
so $M_0^\dagger M_0=v^2\mathbf1$. A symmetry transformation preserves the representative $M_0=v\mathbf1$ precisely when $L=R$, giving
$$
U(N)_L\times U(N)_R\longrightarrow U(N)_{\rm diag}.
$$
The number of broken generators is $2N^2-N^2=N^2$, so the <Goldstone theorem> predicts $N^2$ modes, each a <Goldstone boson>.
Solved by gpt-5.6-sol high.
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