On the one hand, the Improved TSLDA eliminates the singular matrix [S.sub.w] and [S.sub.b] using an approximate matrix method to reduce the time complexity, where it approximately computes the inverse of original eigenvalue matrix with a reverse eigenvalue matrix.

Thus, A(x) is a singular matrix. If n > m, we show that A(x) is a positive semidefinite matrix.

As a result we get the left singular matrix [U.sub.mxr], the diagonal matrix [[summation].sub.rxr], and the right singular matrix [V.sub.nxr].

The reason for using [[sigma].bar] is the interpretation of the smallest singular value of a matrix as the distance between the matrix and the nearest singular matrix, since this is precisely the concept needed to determine the nearness of a stable transfer function to an unstable one.

Without the loss of generality, if matrix D is the near singular matrix, then the matrix's minimum eigenvalue [[lambda].sub.mm] [right arrow] 0 .

However, lansvd can fail for matrices that are nearly rank deficient (problems 18 and 19, marked by "-") because of the inversion of a singular or nearly singular matrix R.

E, A, [[??].sub.i], [B.sub.j], and C are known constant system matrices of appropriate dimensions and E is a singular matrix. The corresponding state time delay [[tau].sub.s,i], i = 1, 2, ..., [N.sub.1], input time delay [[tau].sub.i,j], j = 1, 2, ..., [N.sub.2], and output time delay [[tau].sub.o] are assumed to be known.

From (17), the controllers [u.sub.i] are dependent not only on the coupled matrix B, but also on the singular matrix E.

If the control problem comes from an ordinary differential equation, then E = I and if it comes from a differential-algebraic equation, then E is a singular matrix.