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The scalar triple product la, b, c] equals the signed volume of the parallelepiped defined by a, b, and c (Fig. 1b); the volume is positive if the three vectors are a right-handed system in that order and negative if they are a left-handed system. , if they all lie on a common plane. We can also write la, b,c[ = (a x b , c ) = (b x c , a ) = (c x a , b ) . 32) Since [a, b, a x b[ - lid x bll 2, th~ vector product a • b is oriented, if it is not O, in such a way that {a, b, a x b} form a right-handed system (Fig.

3. Linear Systems and Optimization 49 solution is given in the form of eq. 127). However, if the elements of the matrix A and the components of the vector b are supplied by a physical measurement, all the m equations may be independent because of noise. As a result, eq. 126) may become ill-conditioned or inconsistent. In such a case, a well-conditioned equation that gives a good approximation to x is obtained by "projecting" both sides of eq. 126) onto the eigenspace of A defined by the largest r singular values.

For an (nn)-matrix A, its square root ~ is also symmetric (see eq. 73)). We can also write A observations, we conclude the following: v/-A Tv/A. From these 9 Matrix A is positive semi-definite if and only if there exists a matrix B such that A - B T B . 9 Matrix A is positive definite if and only if there exists a nonsingular matrix B such that A - B TB. 9 If A is a positive semi-definite (nn)-matrix, matrix B T A B is a positive semi-definite ( m m ) - m a t r i x for any n m - m a t r i x B. ~ Nonsingulargeneralized eigenvalue problem Let A be an (nn)-matrix, and G a positive semi-definite (nn)-matrix.