矩阵类的唯一构造器

IF 0.8 Q2 MATHEMATICS

Special Matrices Pub Date : 2019-04-25 DOI:10.1515/spma-2020-0122

T. Hurley

{"title":"矩阵类的唯一构造器","authors":"T. Hurley","doi":"10.1515/spma-2020-0122","DOIUrl":null,"url":null,"abstract":"Abstract Basic matrices are defined which provide unique building blocks for the class of normal matrices which include the classes of unitary and Hermitian matrices. Unique builders for quantum logic gates are hence derived as a quantum logic gates is represented by, or is said to be, a unitary matrix. An efficient algorithm for expressing an idempotent as a unique sum of rank 1 idempotents with increasing initial zeros is derived. This is used to derive a unique form for mixed matrices. A number of (further) applications are given: for example (i) U is a symmetric unitary matrix if and only if it has the form I − 2E for a symmetric idempotent E, (ii) a formula for the pseudo inverse in terms of basic matrices is derived. Examples for various uses are readily available.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"52 - 65"},"PeriodicalIF":0.8000,"publicationDate":"2019-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0122","citationCount":"3","resultStr":"{\"title\":\"Unique builders for classes of matrices\",\"authors\":\"T. Hurley\",\"doi\":\"10.1515/spma-2020-0122\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Basic matrices are defined which provide unique building blocks for the class of normal matrices which include the classes of unitary and Hermitian matrices. Unique builders for quantum logic gates are hence derived as a quantum logic gates is represented by, or is said to be, a unitary matrix. An efficient algorithm for expressing an idempotent as a unique sum of rank 1 idempotents with increasing initial zeros is derived. This is used to derive a unique form for mixed matrices. A number of (further) applications are given: for example (i) U is a symmetric unitary matrix if and only if it has the form I − 2E for a symmetric idempotent E, (ii) a formula for the pseudo inverse in terms of basic matrices is derived. Examples for various uses are readily available.\",\"PeriodicalId\":43276,\"journal\":{\"name\":\"Special Matrices\",\"volume\":\"9 1\",\"pages\":\"52 - 65\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2019-04-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1515/spma-2020-0122\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Special Matrices\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1515/spma-2020-0122\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Special Matrices","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/spma-2020-0122","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 3

摘要

定义了基本矩阵，它为包括酉矩阵和埃尔米特矩阵类的正规矩阵类提供了唯一的构造块。量子逻辑门的独特构建器因此被推导出来，因为量子逻辑门由酉矩阵表示，或者说是酉矩阵。导出了一种将幂等元表示为秩为1的幂等元与增加的初始零的唯一和的有效算法。这用于推导混合矩阵的唯一形式。给出了许多（进一步的）应用：例如（i）U是对称酉矩阵，当且仅当它对于对称幂等元E具有i−2E形式，（ii）导出了关于基本矩阵的伪逆的公式。各种用途的例子都很容易获得。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Unique builders for classes of matrices

Abstract Basic matrices are defined which provide unique building blocks for the class of normal matrices which include the classes of unitary and Hermitian matrices. Unique builders for quantum logic gates are hence derived as a quantum logic gates is represented by, or is said to be, a unitary matrix. An efficient algorithm for expressing an idempotent as a unique sum of rank 1 idempotents with increasing initial zeros is derived. This is used to derive a unique form for mixed matrices. A number of (further) applications are given: for example (i) U is a symmetric unitary matrix if and only if it has the form I − 2E for a symmetric idempotent E, (ii) a formula for the pseudo inverse in terms of basic matrices is derived. Examples for various uses are readily available.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Special Matrices MATHEMATICS-

CiteScore

1.10

自引率

20.00%

发文量

审稿时长

8 weeks

期刊介绍： Special Matrices publishes original articles of wide significance and originality in all areas of research involving structured matrices present in various branches of pure and applied mathematics and their noteworthy applications in physics, engineering, and other sciences. Special Matrices provides a hub for all researchers working across structured matrices to present their discoveries, and to be a forum for the discussion of the important issues in this vibrant area of matrix theory. Special Matrices brings together in one place major contributions to structured matrices and their applications. All the manuscripts are considered by originality, scientific importance and interest to a general mathematical audience. The journal also provides secure archiving by De Gruyter and the independent archiving service Portico.