By Ali H. Sayed

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Likewise, we can verify that the product L s L ; * is upper-triangular. 6) for some diagonal matrix D. ant to show that D is the identity matrix. 4) entries of L1 and LZ must coincide so that the leading entry of D must be unity. 2 also coincide. Hence, the second entry of D is also unity. Proceeding in this fashion we get D = I. l (Evaluation of Cholesky factors) While we obtained the Cholesky factorization of a positivedefinite matrix by performing a sequence of successive Schur complements, this need not be the preferred method numerically for the evaluation of the Cholesky factor.

2 op > 0. If C has p nonzero diagonal entries then A has rank p . The columns of U and V are called the left- and right-singular vectors of A , respectively. Constructive Proof of the SVD One proof of the SVD decomposition follows from the eigen-decomposition of a Hermitian nonnegativedefinite matrix. The argument given here assumes n m, but it can be adjusted to handle the case < n 2 m. This representation simply corresponds to the eigen-decomposition of AA'. The diagonal entries of C2 are the eigenvalues of AA*, which are nonnegative (and, hence, the notation C2); they are also equal to the nonzero eigenvalues of A*A.

The dependency between two real-valued random variables {a:, y} is characterized by their joint probability density function (pdf). Thus, let fm,v (z1y) denote the joint (continuous) pdf of z and y; this function allows us to evaluate probabilities of events of the form: P(a 5 2 5 b , c < 2/ 5 d)=~d~bf=,v(ZIY)dZdY namely, the probability that z and y assume values inside the intervals [a,b] and [c,4 , respectively. Let also fmlv(zly) denote the conditional pdf of z given y; this function allows us to evaluate probabilities of events of the form P(a 5 2 5 b I 9 = Y) = lb f=lv(ZlY)dZ namely, the probability that a: assumes values inside the interval [a, b] given that y is fixed at the value y.