| Summary: | In this paper, we present a new concept of the generalized core orthogonality (called the C-S orthogonality) for two generalized core invertible matrices <i>A</i> and <i>B</i>. <i>A</i> is said to be C-S orthogonal to <i>B</i> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>A</mi><mrow><menclose notation="circle"><mi mathvariant="normal">S</mi></menclose></mrow></msup><mi>B</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><msup><mi>A</mi><mrow><menclose notation="circle"><mi mathvariant="normal">S</mi></menclose></mrow></msup><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>A</mi><mrow><menclose notation="circle"><mi mathvariant="normal">S</mi></menclose></mrow></msup></semantics></math></inline-formula> is the generalized core inverse of <i>A</i>. The characterizations of C-S orthogonal matrices and the C-S additivity are also provided. And, the connection between the C-S orthogonality and C-S partial order has been given using their canonical form. Moreover, the concept of the strongly C-S orthogonality is defined and characterized.
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