A Sherman-Morrison-Woodbury Identity for Rank Augmenting Matrices with Application to Centering
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Matrices of the form A + (V_1 + W_1)G(V_2 + W_2)^* are considered where A is a singular ℓ×ℓ matrix and G is a nonsingular k × k matrix, k <ℓ. Let the columns of V_1 be in the column space of A and the columns of W_1 be orthogonal to A. Similarly, let the columns of V_2 be in the column space of A^* and the columns of W_2 be orthogonal to A^*. An explicit expression for the inverse is given, provided that W_i^* W_i has rank k. W_2 have the same column space. An application to centering covariance matrices about the mean is given.
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