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In this paper, we develop local expansions for the ratio of the centeredmatrix-variate $T$ density to the centered matrix-variate normal density withthe same covariances. The approximations are used to derive upper bounds onseveral probability metrics (such as the total variation and Hellingerdistance) between the corresponding induced measures. This work extends some ofthe results of Shafiei & Saberali (2015) and Ouimet (2022) for the univariateStudent distribution to the matrix-variate setting.

Local normal approximations and probability metric bounds for the matrix-variate $T$ distribution and its application to Hotelling's $T$ statistic