Algebraic Characterizations of Relationships between Different Linear Matrix Functions
Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><msub><mi...
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/3/756 |
| _version_ | 1797623830373466112 |
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| author | Yongge Tian Ruixia Yuan |
| author_facet | Yongge Tian Ruixia Yuan |
| author_sort | Yongge Tian |
| collection | DOAJ |
| description | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><msub><mi>X</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>X</mi><mi>k</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> be a matrix function over the field of complex numbers, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><msub><mi>X</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>X</mi><mi>k</mi></msub></mrow></semantics></math></inline-formula> are a family of matrices with variable entries. The purpose of this paper is to propose and investigate the relationships between certain linear matrix functions that regularly appear in matrix theory and its applications. We shall derive a series of meaningful, necessary, and sufficient conditions for the collections of values of two given matrix functions to be equal through the cogent use of some highly selective formulas and facts regarding ranks, ranges, and generalized inverses of block matrix operations. As applications, we discuss some concrete topics concerning the algebraic connections between general solutions of a given linear matrix equation and its reduced equations. |
| first_indexed | 2024-03-11T09:34:23Z |
| format | Article |
| id | doaj.art-5ff0b9715b8f425b9f5882ae26ac661f |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T09:34:23Z |
| publishDate | 2023-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-5ff0b9715b8f425b9f5882ae26ac661f2023-11-16T17:24:07ZengMDPI AGMathematics2227-73902023-02-0111375610.3390/math11030756Algebraic Characterizations of Relationships between Different Linear Matrix FunctionsYongge Tian0Ruixia Yuan1Shanghai Business School, College of Business and Economics, Shanghai 201400, ChinaShanghai Business School, College of Business and Economics, Shanghai 201400, ChinaLet <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><msub><mi>X</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>X</mi><mi>k</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> be a matrix function over the field of complex numbers, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><msub><mi>X</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>X</mi><mi>k</mi></msub></mrow></semantics></math></inline-formula> are a family of matrices with variable entries. The purpose of this paper is to propose and investigate the relationships between certain linear matrix functions that regularly appear in matrix theory and its applications. We shall derive a series of meaningful, necessary, and sufficient conditions for the collections of values of two given matrix functions to be equal through the cogent use of some highly selective formulas and facts regarding ranks, ranges, and generalized inverses of block matrix operations. As applications, we discuss some concrete topics concerning the algebraic connections between general solutions of a given linear matrix equation and its reduced equations.https://www.mdpi.com/2227-7390/11/3/756block matrixgeneral solutiongeneralized inversematrix equationmatrix expressionrange |
| spellingShingle | Yongge Tian Ruixia Yuan Algebraic Characterizations of Relationships between Different Linear Matrix Functions Mathematics block matrix general solution generalized inverse matrix equation matrix expression range |
| title | Algebraic Characterizations of Relationships between Different Linear Matrix Functions |
| title_full | Algebraic Characterizations of Relationships between Different Linear Matrix Functions |
| title_fullStr | Algebraic Characterizations of Relationships between Different Linear Matrix Functions |
| title_full_unstemmed | Algebraic Characterizations of Relationships between Different Linear Matrix Functions |
| title_short | Algebraic Characterizations of Relationships between Different Linear Matrix Functions |
| title_sort | algebraic characterizations of relationships between different linear matrix functions |
| topic | block matrix general solution generalized inverse matrix equation matrix expression range |
| url | https://www.mdpi.com/2227-7390/11/3/756 |
| work_keys_str_mv | AT yonggetian algebraiccharacterizationsofrelationshipsbetweendifferentlinearmatrixfunctions AT ruixiayuan algebraiccharacterizationsofrelationshipsbetweendifferentlinearmatrixfunctions |