Inverse of invertible standard multi-companion matrices with applications

The inverse of invertible standard multi-companion matrices will be derived and introduced as a new technique for generation of periodic autoregression models to get the desired spectrum and extract the parameters of the model from it when the information of the standard multi-companion matrices is not enough for the extracting of the parameters of the model. We will find explicit expressions for the generalized eigenvectors of the inverse of invertible standard multicompanion matrices such that each generalized eigenvector depends on the corresponding eigenvalue therefore we obtain a parameterization of the inverse of invertible standard multi-companion matrix through the eigenvalues and these additional quantities. The results can be applied to statistical estimation, simulation and theoretical studies of periodically correlated and multivariate time series in both discreteand continuous-time series.


Introduction
Time series arise as recordings of processes which vary over time. A recording can either be a continuous trace or a set of discrete observations. Boshnakov (2001) generate the matrix F from the spectral parameters and then reconstruct the parameters for the required parameterization of the models. The main idea of the multi-companion method for generation of periodic autoregression models is to generate a multi-companion matrix with the desired spectrum and extract the parameters of the model from it.
The backward leading minors of the upper right block of a d-companion matrix F are denoted by δ j , j = 1, 2, ..., d and the corresponding determinants by ∆ j (F ). Now we can compute the determinant of the d-companion matrix F by det(F ) = (−1) (m+1)d ∆ d (F ).
If we use all of the above notations and symbols we can rewrite F in the new form as blocks to be In this paper, we introduce to the class of multi-companion matrices which is the inverse of invertible standard multi-companion matrices when the information of the standard multi-companion matrices is not enough for the extracting of the parameters of the model. The results can be applied to statistical estimation, simulation and theoretical studies of periodically correlated and multivariate time series in both discrete-and continuous-time series.
Theorem 1.1 The standard multi-companion matrix F is invertible iff F is non singular.
Direct calculation shows that the inverse of an invertible standard multi-companion matrix F is and in the next section we need to study the matrix G = F −1 . The new form of the inverse of invertible standard multi-companion can be written as

Inverse of invertible standard multi-companion matrices
Let M and N in the standard multi-companion matrix F be equal to the left and right upper corner blocks respectively(i.e. M = F 1:d,1:m−d and N = δ d ). We will rewrite new form of the inverse of invertible standard multi-companion in Equation (4), to get The companion matrix C from the matrix G can be written as which has the following properties:

Properties of inverse of invertible standard multi-companion matrices
The standard multi-companion matrices have properties which can be as a generalization of the corresponding properties of companion matrices, multi-companion matrices. Now we will see in the next subsections some important properties of the inverse of invertible standard multicompanion matrices that we need for our works, for more see [3], [7] and [8].

Multiplication by multi-companion matrices
For instance, a d-companion matrix G is non-singular if and only if its lower left block B d×d has a non-zero determinant. Now if d = 1, then G is companion, and the corresponding determinants of B d×d is equal to g m,d , which is a scalar.
Note that for any d-companion matrix it is not always possible of writing it as a product of companion matrices, so the converse is not true even in the invertible case. Actually, we can find a permutation of the non-trivial rows which allows for fully factorization. The next theorem shows what we say.
Theorem 3.4 Let G be an invertible multi-companion matrix of order d, then it can be factored as products of d companion matrices to be as where P is a (row) permutation matrix and A i , i = 1, ..., d, are companion matrices.
We will use these useful properties of multiplication by multi-companion matrices and certain facts about the factorization of multi-companion matrix for a generation matrix, for more details see [1] and [8].

Factoring into companion times multi-companion
We are looking for a companion matrix A for which the following equation holds: Hence, the expanded form of (5) is  And so, we can write the elements of these matrices in one of the following equations We can solved (7) explicitly for a j−d+1 , j = d, ..., m (i.e., a d , ..., a m ), The remaining equations involve operations on parts of the rows of G d−1 . Say, g id ). Now from (6) and (8), we have

Factoring into multi-companion times companion
So, we can write the elements of these matrices in one of the following equations

Eigenvalues and eigenvectors of inverse of invertible standard multi-companion matrices
The Jordan canonical forms of inverse of invertible standard multi-companion matrices provide a way to generate eigenvalues and eigenvectors to construct the matrix G and then extract the parameters of the corresponding PAR model from it. Consider the equation Gx = λx, that relates G to an eigenvalue λ and a corresponding eigenvector x.
The eigenvalues of the m × m matrix G are the roots (zeros) of its characteristic polynomial, where λI − G is the characteristic matrix of G.
If the characteristic polynomial T (λ) has distinct roots, then it can be factorized into a product of m linear factors Also, if T (λ) has some s repeated roots, then it can be factorized as follows Remark 3.5 If λ i is an eigenvalue of a matrix A, then the dimension of the linearly independent eigenspace corresponding to λ i is called the geometric multiplicity of λ i , and is denoted by gm(λ i ).
On the other hand, the number of times λ − λ i that appears as a factor in the characteristic polynomial of A is called the algebraic multiplicity of A, and is denoted by am(λ i ). Note that from linear algebra gm(λ i ) ≤ am(λ i ), for more details see [1].

Diagonalizable multi-companion matrices
Clearly that, a square matrix G is called diagonalizable if there is an invertible matrix P such that P −1 GP is a diagonal matrix; the matrix G is said to diagonalize G. The decomposition of G into the form G = P JP −1 is the Jordan matrix decomposition of G where J is a Jordan canonical form of G and P is a non-singular matrix and its columns are the corresponding eigenvectors of G.
Clearly, G is diagonalizable if and only if the geometric multiplicities of all eigenvalues are coincide with the algebraic multiplicities, i.e. gm(λ i ) = am(λ i ) for every distinct eigenvalue λ i .

Non-diagonalizable multi-companion matrices
It is important to note that if a matrix has all distinct eigenvalues (whether real or complex), then it is diagonalizable; in other words, only matrices with repeated eigenvalues might be non-diagonalizable.
However, this happens when the Jordan matrix J is blockdiagonal, as the following structure where J(λ i ), i = 1, ..., s is called a Jordan segment associated with the eigenvalue λ which is made up of g i = gm(λ i ) Jordan blocks to get Each block is associated with a set of columns of P forming a Jordan chain which is called generalized eigenvectors. The sum of dimensions of all Jordan blocks associated with λ i is equal to am(λ i ). The number of Jordan blocks associated with λ i is equal to gm(λ i ), for more details see [5].

Eigenvector and generalized eigenvector of a d-companion matrix
We can added linearly independent vectors to the eigenvectors in order to complete the basis, when G does not have m linearly independent eigenvectors to form the columns of the matrix P .
Suppose that the geometric multiplicity of the eigenvalue is less than its algebraic multiplicity. Choose a single s × s Jordan block J j (λ), j = 1, ..., g, where g = gm(λ). The block J j (λ) is associated with a set of columns of P . Let P j = [x (1) , x (2) , ..., x (s) ] be the portion of P that correspond to the location of the block J j (λ) in the Jordan matrix J.
There exactly one independent eigenvector for each Jordan block which is the first vector in the portion. The following properties are very important and useful for the eigenvectors of any matrix, for more details see [5] and [6].   Theorem 3.9 The union of chains of generalized eigenvectors of G belonging to distinct eigenvalues is linearly independent.
From the above we may be constructed a transition matrix P from the chains of linearly independent generalized eigenvectors of G, and justifies the invertibility of P .
Let GP = P J, we have .., s, and the two equations can be squeezed into one if we adopt the convention x (0) ≡ 0.
We called x (s) a generalized eigenvector of order s associated with the eigenvalue λ if we find a vector x (s) such that In particular case, if s = 1, then (G − λI)x (1) = 0 and x (1) = 0, which is the definition of an eigenvector.
This means that x (i) , i = 1, ..., s, is a generalized eigenvector of order i of G.

Applications
Here we outline the periodic autoregressive models where the inverse of invertible standard multi-companion matrices appear and discuss how the results about such matrices may be useful. Exposition of specific results requires a lot of background information from time series analysis and will be published elsewhere. We say that the process {X t } is a periodically correlated time series, if where µ t = EX t < ∞ and γ X ( We suppose below for simplicity that µ t = 0. A periodic autoregressive process is a periodically correlated process which satisfies a stochastic difference equation of the form where { t } is an uncorrelated periodic white noise process and normally distributed terms with mean zero and periodic variances σ 2 (t) and φ t,i is the autoregression coefficients.
The usual stationary autoregression model can be obtained from equation (9) by putting d = 1. In that case the parameters of the model do not depend on t and the polynomial 1 − Σφ i z i or its companion matrix can be used to study the process, e.g. its spectrum. In the general case, d > 1, the polynomials φ t (z) = 1 − Σφ t,i z i cannot be used with the same success but a natural generalization exists (see [2]). Let m = max(d, p 1 , ..., p d ), Z t = (X t , X t−1 , ..., X t−m+1 ). Define the companion matrices A t = C[φ t,1 , φ t,2 , ..., φ t,m ], t = 1, ..., d, and the where E t and U t are uncorrelated with Z t−1 and Z t−d respectively. Without loss of information we can take every d-th element of the sequence Z t , e.g. Y t = Z td , t = ..., −1, 0, 1, ... . The process Y t is multivariate stationary AR(1), Equation (10) can be used to give full description of the properties of the periodic autoregressive process {X t }, for more see [2].
The matrix G is an inverse of invertible standard multi-companion F of order d and its decomposition into a product of companion matrices is A d ? · · ·?A 1 and we use it when the information of F is not enough therefore we can get it from (10) by multiply both sides by G −1 to get Some interesting properties of the periodic autoregression model can be derived using the results from Section 3.
Knowledge of the Jordan form of multi-companion matrices provides a way to generate periodic models with specified properties by constructing the matrix G and then deriving the parameters of the model by factorizing G into a product of companion matrices then get the inverse for each one of them or directly take the inverse of G then do the same way. This can be useful for selection of appropriate models in simulation studies and in estimation of restricted models for the purpose of hypothesis testing (for example, to test periodic integration it is necessary to estimate a restricted model with the zero-hypothesis roots on the unit circle). Conditions of this kind are difficult to handle when working with the parameters φ t,i themselves, for more see [4] and [5].
Here is an example. Suppose that we wish to generate a 6 × 6 diagonalizable 4-companion matrix G with the following spectral parameters, maybe with the intention of simulating quarterly time series using the generated model. Periodic auroregression moving average models may be generated by applying the above procedure separately to the generation of the autoregression and moving average parts. Figure 1: Plot of observed of simulated PAR model for n = 1000 with φ1,i, φ2,i, φ3,i and φ4,i which generation of reversed synthetic river flow data that is important in planning, design and operation of water resources systems.

Conclusion
We found explicit expressions for the generalized eigenvectors of the inverse of invertible standard multi-companion matrices such that each generalized eigenvector depends on the corresponding eigenvalue. We will discussed some properties such the other matrices, as the factorization of matrices.
Moreover, we obtained a parametrization of the inverse of invertible standard multi-companion matrix through the eigenvalues and these additional quantities. The number of parameters in this parametrization is equal to the number of non-trivial elements of the inverse of invertible standard multi-companion matrix. The results can be applied to statistical estimation, simulation and theoretical studies of periodically correlated and multivariate time series in both discrete-and continuous-time series.