  This paper studies spectral approximation for a positive semidefinite matrix
in the online setting. It is known in [Cohen et al. APPROX 2016] that we can
construct a spectral approximation of a given $n \times d$ matrix in the online
setting if an additive error is allowed. In this paper, we propose an online
algorithm that avoids an additive error with the same time and space
complexities as the algorithm of Cohen et al., and provides a better upper
bound on the approximation size when a given matrix has small rank. In
addition, we consider the online random order setting where a row of a given
matrix arrives uniformly at random. In this setting, we propose time and space
efficient algorithms to find a spectral approximation. Moreover, we reveal that
a lower bound on the approximation size in the online random order setting is
$\Omega (d \epsilon^{-2} \log n)$, which is larger than the one in the offline
setting by an $\mathrm{O}\left( \log n \right)$ factor.
