Rank Equalities Related to the Generalized Inverses <i>A</i>^{‖(<i>B</i>₁,<i>C</i>₁)}, <i>D</i>^{‖(<i>B</i>₂,<i>C</i>₂)} of Two Matrices <i>A</i> and <i>D</i>

• Main Authors: Wenjie Wang, Sanzhang Xu, Julio Benítez
• Format: Article
• Language: English
• Published: MDPI AG 2019-04-01
• Series: Symmetry
• Subjects: Rank; (<i>B</i>, <i>C</i>)-inverse; inverse along an element
• Online Access: https://www.mdpi.com/2073-8994/11/4/539
• ISSN: 2073-8994

Let <i>A</i> be an <inline-formula> <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>&#215;</mo> <mi>n</mi> </mrow> </semantics> </math> </inline-formula> complex matrix. The <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-inverse <inline-formula> <math display="inline"> <semantics> <msup> <mi>A</mi> <mrow> <mo>∥</mo> <mo>(</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>)</mo> </mrow> </msup> </semantics> </math> </inline-formula> of <i>A</i> was introduced by Drazin in 2012. For given matrices <i>A</i> and <i>B</i>, several rank equalities related to <inline-formula> <math display="inline"> <semantics> <msup> <mi>A</mi> <mrow> <mo>∥</mo> <mo>(</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </msup> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msup> <mi>B</mi> <mrow> <mo>∥</mo> <mo>(</mo> <msub> <mi>B</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>C</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </msup> </semantics> </math> </inline-formula> of <i>A</i> and <i>B</i> are presented. As applications, several rank equalities related to the inverse along an element, the Moore-Penrose inverse, the Drazin inverse, the group inverse and the core inverse are obtained.