Volumes of orthogonal groups and unitary groups

The matrix integral has many applications in diverse fields. This review article begins by presenting detailed key background knowledge about matrix integral. Then the volumes of orthogonal groups and unitary groups are computed, respectively. As a unification, we present Mcdonald's volume formula for a compact Lie group. With this volume formula, one can easily derives the volumes of orthogonal groups and unitary groups. Applications are also presented as well. Specifically, The volume of the set of mixed quantum states is computed by using the volume of unitary group. The volume of a metric ball in unitary group is also computed as well. There are no new results in this article, but only detailed and elementary proofs of existing results. The purpose of the article is pedagogical, and to collect in one place many, if not all, of the quantum information applications of the volumes of orthogonal and unitary groups.

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