Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

G.8 MATRIX INVERSION LEMMA

The following property of matrices, which is known as the Sherman–Morrison–Woodbury formula, is useful for deriving the recursive least-squares (RLS) algorithm in Chapter 11.

Lemma G.1 (Matrix inversion). For square nonsingular matrices B and D:

(G.61)

where matrices U and v are not necessarily square.

Proof. See Problem 11.17.

When and are column vectors, the matrix inversion lemma simplifies to the Sherman–Morrison formula:

(G.62)

Since is necessarily a scalar, this case is considerably simpler because of the division by scalar d−1 + vT B−1u. This is the form used to derive the RLS algorithm.