  The radiative decay of $\Lambda (1405)$ is investigated from the viewpoint of
compositeness, which corresponds to the amount of two-body states composing
resonances as well as bound states. For a $\bar{K}N (I=0)$ bound state without
couplings to other channels, we establish a relation between the radiative
decay width and the compositeness. Especially the radiative decay width of the
bound state is proportional to the compositeness. Applying the formulation to
$\Lambda (1405)$, we observe that the decay to $\Lambda \gamma$ is dominated by
the $K^{-}p$ component inside $\Lambda (1405)$, because in this decay $\pi ^{+}
\Sigma ^{-}$ and $\pi ^{-} \Sigma ^{+}$ strongly cancel each other and the $\pi
\Sigma$ component can contribute to the $\Lambda \gamma$ decay only through the
slight isospin breaking. This means that the decay $\Lambda (1405) \to \Lambda
\gamma$ is suitable for the study of the $\bar{K} N$ component in $\Lambda
(1405)$. Fixing the $\Lambda (1405)$-$\pi \Sigma$ coupling constant from the
usual decay of $\Lambda (1405) \to \pi \Sigma$, we show a relation between the
absolute value of the $\bar{K} N$ compositeness for $\Lambda (1405)$ and the
radiative decay width of $\Lambda (1405) \to \Lambda \gamma$ and $\Sigma ^{0}
\gamma$, and we find that large decay width to $\Lambda \gamma$ implies large
$\bar{K}N$ compositeness for $\Lambda (1405)$. By using the "experimental" data
on the radiative decay widths, which is based on an isobar model fitting of the
$K^{-}p$ atom data, we estimate the $\bar{K}N$ compositeness for $\Lambda
(1405)$. We also discuss the pole position dependence of our relation on the
$\Lambda (1405)$ radiative decay width and the effects of the two-pole
structure for $\Lambda (1405)$.
