A note on Johansen's rank conditions and the Jordan form of a matrix

IF 1 4区数学 Q3 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Journal of Time Series Analysis Pub Date : 2024-10-28 DOI:10.1111/jtsa.12789

Massimo Franchi

{"title":"A note on Johansen's rank conditions and the Jordan form of a matrix","authors":"Massimo Franchi","doi":"10.1111/jtsa.12789","DOIUrl":null,"url":null,"abstract":"This note presents insights on the Jordan structure of a matrix which are derived from an extension of the <math>\n <mrow>\n <mi>I</mi>\n <mo>(</mo>\n <mn>1</mn>\n <mo>)</mo>\n </mrow></math> and <math>\n <mrow>\n <mi>I</mi>\n <mo>(</mo>\n <mn>2</mn>\n <mo>)</mo>\n </mrow></math> conditions in Johansen (1996). It is first observed that these conditions not only characterize, as it is well known, the size (1 or 2) of the largest Jordan block in the Jordan form of the companion matrix but more generally the geometric multiplicities, the algebraic multiplicities and the whole Jordan structure for eigenvalues of index 1 or 2. In the context of the Granger representation theorem, this means that the Johansen rank conditions do more than determine the order of integration of the process. It is then shown that an extension of these conditions leads to the characterization of the Jordan structure of any matrix.","PeriodicalId":49973,"journal":{"name":"Journal of Time Series Analysis","volume":"46 4","pages":"796-805"},"PeriodicalIF":1.0000,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/jtsa.12789","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Time Series Analysis","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1111/jtsa.12789","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}

引用次数: 0

Abstract

This note presents insights on the Jordan structure of a matrix which are derived from an extension of the $I (1)$ and $I (2)$ conditions in Johansen (1996). It is first observed that these conditions not only characterize, as it is well known, the size (1 or 2) of the largest Jordan block in the Jordan form of the companion matrix but more generally the geometric multiplicities, the algebraic multiplicities and the whole Jordan structure for eigenvalues of index 1 or 2. In the context of the Granger representation theorem, this means that the Johansen rank conditions do more than determine the order of integration of the process. It is then shown that an extension of these conditions leads to the characterization of the Jordan structure of any matrix.

查看原文本刊更多论文

关于Johansen秩条件和矩阵的Jordan形式的注记

本文介绍了从Johansen（1996）中I(1)和I(2)条件的扩展中导出的矩阵的约旦结构的见解。首先观察到，这些条件不仅表征了伴随矩阵的约旦形式中最大约旦块的大小（1或2），而且更一般地表征了指标1或2的特征值的几何多重性、代数多重性和整个约旦结构。在格兰杰表示定理的背景下，这意味着约翰森秩条件不仅仅决定了过程的积分顺序。然后证明了这些条件的扩展导致任何矩阵的约旦结构的表征。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Journal of Time Series Analysis 数学-数学跨学科应用

CiteScore

2.00

自引率

0.00%

发文量

审稿时长

6-12 weeks

期刊介绍： During the last 30 years Time Series Analysis has become one of the most important and widely used branches of Mathematical Statistics. Its fields of application range from neurophysiology to astrophysics and it covers such well-known areas as economic forecasting, study of biological data, control systems, signal processing and communications and vibrations engineering. The Journal of Time Series Analysis started in 1980, has since become the leading journal in its field, publishing papers on both fundamental theory and applications, as well as review papers dealing with recent advances in major areas of the subject and short communications on theoretical developments. The editorial board consists of many of the world''s leading experts in Time Series Analysis.