On generalized inverses of a block in a partitioned matrix

AbstractGiven symmetric generalized inverses of Aχx′c, where χ is a vector with n components, we obtain formulae for the corresponding generalized inverses (symmetric) of A. An application of such formulae in linear models is suggested

Clamping effect on the piezoelectric responses of screen-printed low temperature PZT/Polymer films on flexible substrates

This paper introduces a new flexible lead zirconate titanate (PZT)/polymer composite material that can be screen-printed onto fabrics and flexible substrates, and investigates the clamping effect of these substrates on the characterization of the piezoelectric material. Experimental results showed that the optimum blend of PZT/polymer binder with a weight ratio of 12:1 provides a dielectric constant of 146. The measured value of the piezoelectric coefficient d33 was found to depend on the substrate used. Measured d33clp values of 70, 40, 36 pC N−1 were obtained from the optimum formulation printed on Polyester–cotton with an interface layer, Kapton and alumina substrates, respectively. The variation in the measured d33clp values occurs because of the effect of the mechanical boundary conditions of the substrate. The piezoelectric film is mechanically bonded to the surface of the substrate and this constrains the film in the plane of the substrate (the 1-direction). This constraint means that the perpendicular forces (applied in the 3-direction) used to measure d33 introduce a strain in the 1-direction that produces a charge of the opposite polarity to that induced by the d33 effect. This is due to the negative sign of the d31 coefficient and has the effect of reducing the measured d33 value. Theoretical and experimental investigations confirm a reduction of 13%, 50% and 55% in the estimated freestanding d33fs values (80 pC N−1) on Polyester–cotton, Kapton and alumina substrates, respectively. These results demonstrate the effect of the boundary conditions of the substrate/PZT interface on the piezoelectric response of the PZT/polymer film and in particular the reduced effect of fabric substrates due to their lowered stiffness

On a theorem of Rao on g-inverses of matrices

Consider matrices Ai(m × ni)i =1(1)k and let Λ(m × m) be a p.d. matix. Rao(1967) shows that if Σ R(Ai)=m, Λ-1=Σ Ai(A•iΛ Ai)-A•i ⇔ A•i Λ Aj =0 for each i, j, i ≠ j. The present note gives an interesting generalisation of this theorem of Rao

Generalized inverses of partitioned matrices and recalculation of least squares estimates for data or model changes

Let A be a m × n matrix and a be an m × 1 vector. In this paper, the authors have obtained formulae for computing several types of g-inverses of (A: α) from the corresponding g-inverses of A. Similar formulae are obtained for computing g-inverses of A from those of (A: α). These formulae are useful for revising least squares estimates in the following situations: (i) deletion or addition of an observation; and (ii) deletion or addition of a parameter in the linear model. Finally, the paper also gives formulae for computing g-inverses of A+BDC* from those of A when D is nonsingular

Testing for the mean vector of a multivariate normal distribution with a possibly singular dispersion matrix and related results

Let y [approximate] p ([mu], [Sigma]) where [Sigma], possibly singluar, is unknown. We develop a test for H0: [mu] = [mu]0 against H1:[mu][not equal to][mu]0 based on a random random sample fromy as an extension of Hotelling's T2 test. We show that the same procedure can be used to compute the test statistic in both the cases of the singular and the positive definite dispersion matrices. This development involves the singular Wishart distribution. Motivated by Khatri (1968), we also obtain an expression for the density of the Wishart distribution p(k, [Sigma]) in each o the following cases: (i)[Sigma] positive definite, kWishart distribution singular multivariate distributions

Determination of a matrix by its subclasses of generalized inverses

The main result of the paper is: AB+A=A and BA+B=B ⇒ A=B where A+ and B+ are the unique Moore-Penrose inverses of A and B respectively. This is a stronger result than uniqueness of Moore-Penrose inverse. Some other results are established generalizing the earlier result of Rao and Mitra (1971) that a matrix is uniquely determined by the entire class of its g-inverses