On the Minimum Polynomial and Applications

In this note we study the computation of the minimum polynomial of a matrix $A$ and how we can use it for the computation of the matrix $A^n$. We also describe the form of the elements of the matrix $A^{-n}$ and we will see that it is closely related with the computation of the Drazin generalized inverse of $A$. Next we study the computation of the exponential matrix and finally we give a simple proof of the Leverrier - Faddeev algorithm for the computation of the characteristic polynomial.

Experimental research of the angular superresolution of correlated signal sourcesby the minimum polynomial method.

Experimental results on the super-resolution of two closely spaced signal sources obtained using a car radar of the millimeter wavelength range are presented. The peculiarities of the experimental conditions were a high mutual correlation of signal sources and an extremely short input process, which consisted of only one sample. The estimation of the number of sources was carried out using the method of the minimum polynomial of the correlation matrix of signals in the antenna array, which provided the probability of correct estimation of the number of sources equal to 100% in all scenarios of the signal situation. To find the angular coordinates of the sources, we used the method of scanning the AA beam, the spectral and root methods of the minimum polynomial. Comparative analysis showed that the root method is more efficient and provides resolution of sources in cases of smaller angular distance between them.

The method of forming virtual receiving channels in the automobile MIMO-radar

The article is intended for specialists in the field of radar, radio engineering and telecommunications. It considers the problem of forming virtual receiving channels in a car MIMO radar to increase the antenna aperture in the horizontal plane. The case is investigated when two cars fall into the main beam of the radar antenna pattern, and therefore, the application of super-resolution methods in azimuth is required. In modern vehicles, this option is required for vehicle collision avoidance and driver assistance systems. The use of MIMO technology makes it possible to form a larger antenna array (AR) for reception, the so-called virtual AR. This becomes possible due to a special choice of the topologies of the location of the transmitting and receiving channels, as well as due to the multiplication of the probing signals in each transmitting channel by an individual code. As a result, the resolving power of the AR increases in the direction finding of the target. A radar with “short” sounding chirp pulses is considered, in which the range is measured by the frequency method. The parameters of the signal in the radar are chosen so that the contribution of the terms associated with the speed of the target on the pulse duration to the beat frequency on the receiving side is negligible. Analytical expressions are obtained for the received signals under conditions of the Doppler frequency shift and “short” probing pulses. A method for generating signals in virtual receiving channels using an encoder only on the transmitting side and without a decoder on the receiving side is proposed. The use of various codes is investigated and their effectiveness is compared. For direction finding of targets, the method of the minimum polynomial of the correlation matrix of the received signals is applied. The method involves assessing the degree of the minimum polynomial of the correlation matrix of the input process in the AR based on a statistically valid root-mean-square criterion. This method allows adaptive estimation of the number of signal sources and has a super-resolution function. In contrast to the known works, the main attention is paid to the case of a short sample of the input process, when the number of samples is less than the dimension of the virtual AR. In this case, the sample correlation matrix is degenerate. The results of numerical modeling are given for the accuracy of azimuth measurement and the probability of correct resolution of two targets. A natural experiment was carried out. The presented results demonstrate the efficiency of the proposed concept, high accuracy of azimuth measurements, and the possibility of super resolution of two targets in the case of a short sample.

ALGORITMA POLINOMIAL MINIMUM UNTUK MEMBENTUK MATRIKS DIAGONAL DARI MATRIKS PERSEGI

In mathematics, matrices have many uses, they are finding solutions of a linear equation system, looking for specific solutions of differential equations, determining state classification on Markov chains, and so on. There is a special matrix in matrix theory, that is a diagonal matrix. The diagonal matrix is a matrix whose all non-diagonal entries are primarily zero so that the product of the diagonal matrix can be computed by considering only the components along the main diagonal. A square matrix can sometimes be formed into a diagonal matrix. If a non-diagonal square matrix A can be conjugated with a diagonal matrix, then there is an invertible matrix P so PAP-1=D, where D is a diagonal matrix and P is said to diagonalize A. To find a square matrix diagonalizable or not, many researchers usually use eigenvalues and eigenvectors evaluation. In this study, we discuss that the other way to form a diagonal matrix by using Minimum Polynomial Algorithm.

Transverse Hilbert schemes and completely integrable systems

In this paper we consider a special class of completely integrable systems that arise as transverse Hilbert schemes of d points of a complex symplectic surface S projecting onto ℂ via a surjective map p which is a submersion outside a discrete subset of S. We explicitly endow the transverse Hilbert scheme Sp[d] with a symplectic form and an endomorphism A of its tangent space with 2-dimensional eigenspaces and such that its characteristic polynomial is the square of its minimum polynomial and show it has the maximal number of commuting Hamiltonians.We then provide the inverse construction, starting from a 2ddimensional holomorphic integrable system W which has an endomorphism A: TW → TW satisfying the above properties and recover our initial surface S with W ≌ Sp[d].