Spectrum of a matrix

About: Spectrum of a matrix is a research topic. Over the lifetime, 1064 publications have been published within this topic receiving 19841 citations. The topic is also known as: matrix spectrum.

The use of polynomials

Abstract: Publisher Summary If the eigenvalues are known approximately, the convergence of the power method can largely be improved by increasing the quotient |λ 1 /λ 2 |. The most simple case is that of adding to all eigenvalues the same constant c . If all eigenvalues are positive, one subtracts (λ 2 + λ n )/2. Knowing for instance that a four-rowed matrix has approximately the eigenvalues—12; 6; 4; 1, one subtracts 3,5 and has the new approximate eigenvalues—8,5; 2,5; 0,5; − 2, 5. Therefore, the quotient λ 1 /λ 2 has increased from 12/6 = 2 to 8,5/2,5 = 3,4. As is well known the eigenvectors remain unchanged. When λ 1 > 0, but the other eigenvalues are partially 2 and the first negative eigenvalue from all eigenvalues. In many cases, this is the best choice and in every case, it will improve the relation. This chapter discusses the use of polynomials and quadratic transformations.

‘jumps’ in macroeconomic models: an example when eigenvalues are complex-valued*

Abstract: The dynamic properties of continuous-time macroeconomic models are typically characterised by having a combination of stable and unstable eigenvalues. In a seminal paper, Blanchard and Kahn showed that, for linear models, in order to ensure a unique solution, the number of discontinuous or ‘jump’ variables must equal the number of unstable eigenvalues in the economy. Assuming no zero eigenvalues and that all eigenvalues are distinct, this also means that the number of predetermined variables, otherwise referred to as continuous or non- ‘jump’ variables, must equal the number of stable eigenvalues. In this paper, we investigate the application of the Blanchard and Kahn results and establish that these results also carry through for linear dynamical systems where some of the eigenvalues are complex-valued. An example with just one complex conjugate pair of stable eigenvalues is presented. The Appendix contains a general n-dimensional model.

Spectral analysis of finite difference models of open structures in time and frequency domain

Abstract: We present an analysis of extended, FDTD-like methods which is based on the eigenvalues of the iteration matrix. For a stable update scheme these eigenvalues must lie within the unit circle, and this kind of analysis is applied to systems with radiation losses. For small models, it is possible to compute the eigenvalues numerically and to compare them to the spectrum of the corresponding time-continuous formulation. This allows to assess the accuracy and stability properties of different implementation variants.