By Graham M. L. Gladwell, Antonino Morassi
The papers during this quantity current an summary of the final facets and useful purposes of dynamic inverse equipment, throughout the interplay of numerous issues, starting from classical and complicated inverse difficulties in vibration, isospectral structures, dynamic tools for structural identity, lively vibration keep watch over and harm detection, imaging shear stiffness in organic tissues, wave propagation, to computational and experimental points correct for engineering difficulties.
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Extra resources for Dynamical Inverse Problems: Theory and Application (CISM International Centre for Mechanical Sciences)
Linear Algebra and Its Applications, 271:257–272, 1998. L. Gladwell and O. Rojo. Isospectral ﬂows that preserve matrix structure. Linear Algebra and Its Applications, 421:85–96, 2007. L. R. Lawrence, and D. Siegel. Isospectral ﬁnite element membranes. Mechanical Systems and Signal Processing, 23:1986–1999, 2009. H. Golub. Some uses of the Lanczos algorithm in numerical linear algebra. H. Miller, editor, Topics in Numerical Analysis. Academic Press, 1973. T. Nanda. Diﬀerential equations and the QR algorithm.
K ˜ n−1 , then K ˜ n−1 is full. ˜ 2, K are non-negative. If we form powers, K n−1 ˜ ˜ It can be shown that K is PD, so that K is O, and K is SO, and K−1 is O. Total positivity is linked to staircase form through the result: If A ∈ Sn is NTN, it is a staircase matrix. Matrix Inverse Eigenvalue Problems 23 Gladwell (1998) proved that if A ∈ Sn , if P denotes one of the properties NTN, O, TP, and if B is related to A by equations (14), (15) with μ not an eigenvalue of A, then B has property P iﬀ A has property P .
This means that, given A, B, we may pass from A to B in n − 1 steps Gμj , and the order in which we take these steps is immaterial. We used the equation RA = BR to show that if A is a Jacobi matrix, then so is B. There is a more general result relating to a so-called staircase matrix. A symmetric staircase matrix has its non-zero entries clustered around the diagonal as in a staircase, as shown in Figure 4. It is easily shown that if A ∈ Sn is a staircase matrix, and B = Gμ A, then B is a staircase matrix with the same pattern.