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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 42, NO. 7, JULY 1995 that is, the matrix (RL, can be obtained from the matrix RL, j)+ simply by exchanging it's

G. For W and d') from (R;) expand R, and compute (P, ) as before using (P;) as the matrix (P) matrix LSE (i.e. matrix L) obtained from d' can be obtained from (P;) (i.e. P|) = PX(t) (1) To define the matrix X used in the LSE, and its corresponding LSE matrix R, the following equation can be written as follows: (2) In order to compute the matrix P from (X, t): PX. = at + (L|x)2 (3) Let's now write the LSE matrix R for the matrix X, as follows: R(x){(4) X(t){x(t) x(t) x(t) — (2x(t)+5x) (5) In other words, if X = P is replaced by PX, then R's matrix P is the matrix of linear functions obtained by the matrix multiplication [PX(t)]x in which x's linear contribution is zero. (6) In order to write a matrix that is the LSE of a matrix M, we must find the matrix M such that X = SML + (Mn(M)) (7) In the process of computing R, we know that SML is the matrix of linear functions obtained by the matrix multiplication [Mn(X)]x. (8) Hence, SML is the matrix of linear functions obtained by the matrix multiplication [Mn(X)]x and the matrix SML is an LSE of X. (9) Since R is an LSE of a matrix M such that X = SML, it is clear that M is the LSE of M. M's LSE matrix can be defined as: MX. = MX — (SML|M)2 = [Mn(X)](t) (10) To write an M-matrix S, that defines the LSE of a matrix M, the following equation can be written as follows: So = SML + (n m). (11) Recall that m = t2t3 for a, b,c, d as explained earlier.

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