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Let H be a Haar distributed random matrix on the group of pxp real orthogonalmatrices. Partition H into four blocks: (1) the (1,1) element, (2)the rest ofthe first row, (3) the rest of the first column, and (4)the remaining(p-1)x(p-1) matrix. The marginal distribution of (1) is well known. In thispaper, we give the conditional distribution of (2) and (3) given (1), and theconditional distribution of (4) given (1), (2), (3). This conditionalspecification uniquely determines the Haar distribution. The two conditionaldistributions involve well known probability distributions namely, the uniformdistribution on the unit sphere in p-1 dimensional space and the Haardistribution on (p-2)x(p-2) orthogonal matrices. Our results show how toconstruct the Haar distribution on pxp orthogonal matrices from the Haardistribution on (p-2)x(p-2) orthogonal matrices coupled with the uniformdistribution on the unit sphere in p-1 dimensions.

A decomposition result for the Haar distribution on the orthogonal group